Simple Answer: The Greek Head.
How does one arrive at this answer, and what hermeneutic does one use to arrive at this answer?
Introduction
In order to understand the answer, we must review the hermeneutic associated with this answer as well as discuss why other hermeneutics fail, or are considered "Opinion Based". Let's first consider the authorial intent of the text:(Nestle GNT 1904)
καὶ μίαν ἐκ τῶν κεφαλῶν αὐτοῦ ὡς ἐσφαγμένην εἰς θάνατον, καὶ ἡ πληγὴ
τοῦ θανάτου αὐτοῦ ἐθεραπεύθη. καὶ ἐθαυμάσθη ὅλη ἡ γῆ ὀπίσω τοῦ θηρίου
Interlinear Translation: "And one of the heads of it as having been slain to death; and the wound of death of it was healed; and marveled all the earth after the beast."
In comparing translations, the TR Stephanus 1550 uses "εἶδον" between καὶ and μίαν but this is simply saying "I saw" or "noticed", or "observed", which is restating what vs 1 of the same chapter states; therefore the translators refrained from repeating the statement. The Siniaticus, Alexandrinian, and Vaticanus codexes all agree "καὶ μίαν ἐκ τῶν κεφαλῶν"(one of the heads) is the Original Intent of the text. Irenaeus of Lyons(d.202) affirms this in Adversus Haereses(Against Heresies) in Book V, Chapter 28:2,
"John has thus described in the Apocalypse: And the beast which I had
seen was like a leopard, and his feet as of a bear, and his mouth as
the mouth of a lion; and the dragon conferred his own power upon him,
and his throne, and great might. And one of his heads was as it were
slain unto death; and his deadly wound was healed, and all the world
wondered after the beast."
Therefore, our conclusion is "one of the heads" is the Authorial Intent of this passage.
Which Hermeneutic System of Eschatology best describes the meaning of the text?
The 4 Main Systems of Eschatology are: Idealist, Preterist, Futurist/Dispensationalist and Historicist. It is in these systems of eschatology that the argument of "Opinion Based" is referenced, as 3 of these systems have little or specious support for an answer, or outrightly reject the Authorial Intent, though we have clearly established it.
The Idealist Interpretation is described by Cornelis Venema as:
The idealist approach differs from the first three approaches in its
reluctance to identify any particular historical events, institutions,
or people with the visions of the book of Revelation.
Furthermore, he explains,
"Idealism acknowledges that the book of Revelation was originally
written to encourage the early church in its struggles under religious
and political persecution."(Both sources taken from here).
An Idealist interpretation sees everything as symbolic or allegorical, with no corresponding basis in literal or historical events. An example of an Idealist interpretation of Rev. 13 can be found in Matthew Henry's Commentary on this passage,(Rev. 13:1-3)
The wounding the head may be the abolishing pagan idolatry; and the
healing of the wound, introducing popish idolatry, the same in
substance, only in a new dress, but which as effectually answers the
devil's design.(Taken from here)
There is no attempt to describe this wound except to say "pagan idolatry", and then "popish idolatry" is introduced, though not explained, nor correlated with the text. This interpretation is entirely subjective, hence the accusation of this method of interpretation as being "Opinion Based" is well founded.
The next method of Eschatological Interpretation is the Preterist Interpretation.
The Preterist Interpretation can be described as,
a Christian eschatological view that interprets some (partial
preterism) or all (full preterism) prophecies of the Bible as events
which have already happened. This school of thought interprets the
Book of Daniel as referring to events that happened in the 2nd century
BC, while seeing the prophecies of Revelation as events that happened
in the first century AD. Preterism holds that Ancient Israel finds its
continuation or fulfillment in the Christian church at the destruction
of Jerusalem in AD 70.(from Wikipedia)
A Preterist Interpretation of this passage says,
"The Roman beast with its seven heads represents seven Caesars
beginning with the first, Julius Caesar. The head of the beast that
had a fatal wound in Revelation 13:3 is the sixth head of the beast
and the sixth Caesar, Nero, who committed suicide by stabbing himself
in the neck. After Nero’s death, the Roman Empire collapsed into civil
war. Peace and order was not restored until the rise of the Flavian
Dynasty one year later. Therefore, the Flavian Dynasty beginning with
Caesar Vespasian is the beast whose wound had been healed in
Revelation 13:3. The healing of the wounded head of the beast is a
resurrection motif."(taken from here)
It's a moot point that Nero chickened out and had his aid stab him; the main issue with this system of interpretation is "the beast(Nero)" did in fact die; there was no resurrection. Substituting "Caesar" for "Nero" when your system of interpretation calls for an individual is disingenuous, and as a system of interpretation this pattern is repeated many times. To make "Revelations" fit their model, they must contort and distort the details; the main one being John's writing of Revelations occurred in 95AD, after the Fall of the Temple in Jerusalem. When it says these things will "shortly come to pass", and you use 95AD as your starting point, any previous explanation of events not specifically referred to in the past belong in the future. Therefore, the "Beast" that was, is not, and shall ascend out of the bottomless pit and go to perdition(Rev. 17:8) cannot be Nero, nor Vespasian as both were deceased by the time Revelations was written. But Preterists remain undeterred, and so one can say very loud and arguably that their view is "Opinion Based".
Futurists and Dispensationalists
It is with this interpretive view that more clarity is needed to be focused, because there is an element of truth in waiting for future events to happen.
Futurism is defined as
a Christian eschatological view that interprets portions of the Book
of Revelation and the Book of Daniel as future events in a literal,
physical, apocalyptic, and global context.(Source Wikipedia)
Dispensationalists are Futurists with the distinction that
Biblical history as divided deliberately by God into defined periods
or ages to each of which God has allotted distinctive administrative
principles. According to dispensationalist interpretation, each age of
the plan of God is thus administered in a certain way, and humanity is
held responsible as a steward during that time.(Source Wikipedia )
The Main Point of difference between Futurists and Dispensationalists is the administration(dispensation) of Israel during the Millennial Reign. Since Dispensationalists see the Church Age as distinct from Christ's Literal Reign in Jerusalem, where the Law is fulfilled and Ezekiel's Temple becomes reality, the Church is "separate"; most dispensationalists believe in a pre-tribulation rapture, where the Church is caught up to Heaven, waiting for the Final Judgment, while Israel is alive and present during the Millennial Reign.
Noted Classic Dispensationalist Charles Ryrie has been frequently quoted as saying,
“A dispensationalist keeps Israel and the church distinct... This is
probably the most basic theological test whether or not a man is a
dispensationalist, and it is undoubtedly the most practical and
conclusive. A man who fails to distinguish Israel and the Church will
inevitably not hold to dispensational distinctions; and one who does,
will."(Taken from here)
Since Dispensationalists hold to a Literal, Historical/Grammatical fulfillment of God's Promises to Israel, there can be no confusion as to God's administration of His Promises: the Church is "Raptured", while Israel administers the Final Chapter of God's Administration, with Christ Himself in Jerusalem, overseeing the effort.
Futurists and Dispensationalists have been vindicated in recent years, as a Literal Nation of Israel was reborn in 1948. The promises of God, particularly of Isa. 66:8,(KJV)
Who hath heard such a thing? who hath seen such things? Shall the
earth be made to bring forth in one day? or shall a nation be born at
once? for as soon as Zion travailed, she brought forth her children.
In May 14, 1948, David Ben-Gurion announced to the world the nation of Israel, after the last British troops left. Even more miraculously, the 5 Arab armies which vowed to "wipe Israel off the map" were soundly defeated, though they outnumbered them significantly. The God of Israel clearly aided them in fighting their battles, as was prophesied in the Old Testament.
A Futurist/Dispensational view of the Rev. 13:3 text is rife with problems: John Walvoord writes on this passage,
"The identification of a head with the government over which he has
authority is not a strange situation. The person is often the symbol
of the government, and what is said of the government can be said of
him. Although verse 3 will continue to be a subject of controversy,
the theological reasons for resisting an actual resurrection of a
historical character to head the revived Roman Empire are so great as
to render it improbable even though such personages as Nero and Judas
Iscariot will continue to attract the attention of modern students of
the book of Revelation. The beast is both personal and the empire
itself; so also is the head. The revival of the future empire is
considered a miracle and a demonstration of the power of Satan."(from here)
While Walvoord concedes the "Head" is a kingdom, and not an individual, because of the "forced" literal(historical/grammatical) interpretation of the passage he has to treat the "severed head" like an individual. Other Dispensationalists are not so reluctant; they picture an individual with a massive head wound, who is pronounced dead, yet at the last minute is healed. The Only Problem with this, as Walvoord admits, is it is not the Authorial Intent of the text. There are 7 Heads, it is a 7 Headed Beast. Because Dispensationalists view the entire book of Revelations as "Progressive", they've concluded a "Revived Rome" as the Head, yet the "Beast" was, and is not"(Rev. 17:8). Rome IS, therefore by virtue of this passage it cannot be the "IS NOT". Rome is the 6th Head, there is another Head called "Revived Rome", or the "Holy Roman Empire", which I will explain further on. But it cannot be the "severed Head" by reasons I just described.
The Futurists offer a variety of opinions; generally centered around Rome being the severed Head, so I won't go into detail, but this text ties eschatological knots for Futurist/Dispensationalists so although it is mistakenly "dramatized" in the "Left Behind" series, and other renditions, there is no consensus as to it's meaning; consequently those of this view consider any answer as "Opinion Based".
Historicist Interpretation
The Final Eschatological viewpoint for our consideration is the Historicist one.
The Historicist view is described as:
a method of interpretation of Biblical prophecies, associates symbols
with historical persons, nations or events. It can result in a view of
progressive and continuous fulfillment of prophecy covering the period
from Biblical times to the Second Coming.(Source Wikipedia)
Though this interpretive view has been widely used by numerous luminaries, including John Knox, Martin Luther, John Calvin, and others, it is the interpretive method seen as the most fraught with danger; it's main criticism is that there is no universal agreement as to the outline of church history. Therefore, there is no uniform schema to apply to historical events, and individuals have been free to conjecture whatever events best describe their viewpoint.
A classic example of this is Martin Luther's calling of Pope Leo X, the "Antichrist", in which the same 'compliment' was passed back to Martin Luther by Pope Leo X. Of course, neither individual fits the historical narrative, but that didn't prevent them from using a Historicist Viewpoint for some fancy ecclesial namecalling!
However, this viewpoint best answers the question, "The Revelation of Jesus Christ, which God gave unto him, to shew unto his servants things which must shortly come to pass"(Rev. 1:1 KJV), in that it agrees with events as unfolding in history; not as having "already happened", unless the context of the event proves it already happened, nor waiting for future fulfillment, unless it is clear (Christ's Return) that it is a future event. It allows for the fulfillment of Revelations throughout the spectrum of history, not "bunching everything at the beginning"(Preterism), nor waiting for everything to happen in the end (Futurism/Dispensationalism), or worse, ignoring the events(Idealism). Rather, if properly applied within textual and hermeneutical considerations can give us valuable insight to the gradual revealing of Revelations.
Meanings of the Symbols
Historicism does share commonality with Futurism in that it sees the symbols used in the Book of Daniel to confer meaning in the Book of Revelations. In Dan. 7:2 it says,(KJV)
Daniel spake and said, I saw in my vision by night, and, behold, the
four winds of the heaven strove upon the great sea.
From the Book of Daniel "great sea" is not described; therefore it could lend itself to any variations or bodies of water. Yet in the Book of Revelations we observe the "beast" arise out of the sea, and carries with it many of the descriptive elements seen in Dan. 7. Furthermore, in Rev. 17:15, the angel says,
The waters which thou sawest, where the whore sitteth, are peoples,
and multitudes, and nations, and tongues.(KJV)
We are told earlier in vs 7,
Wherefore didst thou marvel? I will tell thee the mystery of the
woman, and of the beast that carrieth her, which hath the seven heads
and ten horns.(KJV)
so we realize that the woman(Great Harlot) is sitting on the beast which emerged from the sea in Rev. 13:1. The word in Rev. 17:15 used is "ὕδατα"(hydata), which from Thayers can mean
figuratively used of many peoples, Revelation 17:1, as the seer
himself explains it in Revelation 17:15, cf. Nahum 2:8; of a quantity
of water likened to a river.
Although this is not the same word (θαλάσσης-thalasses) used in Rev. 13:1, since the context is clearly figurative and it also describes the word "great sea" used by the LXX in Dan. 7:2. What's more important is the angel gives us the meaning, "peoples, multitudes, nations, tongues", so we are not left to conjecture the meaning within it's symbolic context, in both Daniel and Revelations.
From here we move on to the beasts of Daniel, and then to the Beast of Revelations.
The 1st beast of Daniel was a lion, which stands like a man, and a man's heart is given to it. The next beast is a bear, and the 3rd beast is a leopard. The 4th beast is "dreadful and terrible"; no animal is given to it, because although the 1st 3 beasts are representative of kingdoms, of which likewise is the 4th, no animal 'matches' the description of this beast.
From Dan. 2 we see Daniel describing to Nebuchadnezzar the succession of kingdoms, starting with his that will reign upon the earth. What is important to understand is it is within these specific kingdoms that we are to identify the future unfolding of prophetic history: there are many kingdoms, and there are many kings, each of them not without their significance and many of them also mentioned in the bible. But nations such as Egypt, Assyria, Libya, Ethiopia, etc., are not part of the prophetic revelation in Chapter 2; to place them there anyway is to void the intent of the passage and substitute one's particular understanding against the prophetic intention of the verses.
The order of prophetic revelation, as well as historical importance is Babylon, Media-Persia, Greece, and Rome. These nations directly interact with Israel and it's future; as oppressor "captor" nations, but more importantly those nations who stand in opposition to the Kingdom of God, which was represented physically in Israel, yet spiritually by the church. This is no more evident than in Dan. 2:34, where it says
Thou sawest till that a stone was cut out without hands, which smote
the image upon his feet that were of iron and clay, and brake them to
pieces.(KJV)
The "Stone", the same "stone which the builders rejected" is Christ. These particular kingdoms are allowed to grow, taking their place in historical order until collectively the 'man' which is described reaches his height when the 10 "toe" nations are revealed. It isn't until the statute historically reaches it's zenith that the "stone" smashes the toes, and disintegrates the statue. Therefore, any interpretation that permits the statue to stand has failed to reconcile this event, and thereby can be dismissed.
Reconciling the Kingdoms of Dan. 2 and 7 to the Beasts of Revelations
Most Futurist and Historicist Commentaries agree with the succession of kingdoms in Dan. 2 and 7 as being Babylon, Mede-Persia, Greece, and Rome; in fact the main reason is that they historically succeed one another, and a futurist has no problem with events that have proven to have happened in the past.
In identifying the "Heads" and "Horns" of the Beast of Rev. 13 is where a Historicist and Futurist interpretation parts company; the Main Reason is that a Futurist interpretation must see the Book of Revelations as "successive"; therefore the narrative of Revelations must, in their view, follow a path that only a future interpretation can bring. Therefore, any event recorded in Revelations that appears "out of sequence" can only be interpreted as "waiting to be fulfilled", whereas a historicist can argue that the event occurred, and we must reconcile our understanding to the historical event that took place.
Futurism gets its label from its refusal to see unfulfilled passages
as having been fulfilled by approximately similar events in the past.
Hence, it holds that many of the events in the book of Revelation
await future fulfillment:
The futurist generally believes that all of the visions from
Revelation Rev. 4:1+ to the end of the book are yet to be fulfilled in
the period immediately preceding and following the second advent of
Christ. The reason for the view is found in the comparison of
Revelation Rev. 1:1+, Rev. 1:19+ and Rev. 4:1.(Both sources taken from here).
Historical Interpretation of the Heads
In Rev. 12:5(KJV) it says,
"And she brought forth a man child, who was to rule all nations with a
rod of iron: and her child was caught up unto God, and to his throne."
A Historicist interpretation* establishes the birth of Christ as the event occurring in Rev. 12:5; from this event we can work from this recognized point of history both backwards and forwards. As Historicism covers the broad spectrum of history, and is not isolated in finding meaning in the "beginning", or the "future", we can determine if events described in prophecy were fulfilled, or have yet to be fulfilled.
The beasts of Dan. 7 give us the clues to identify the beasts of Revelations. In Dan. 7, we have previously stated that the "beasts" began with Babylon, then Mede-Persia, then Greece, then Rome. We therefore can rule out Egypt and Assyria since no mention of either of them occurs in Daniel.
What has allowed commentators to include these kingdoms is when the heads are described in Rev. 17:9-11,(KJV)
And here is the mind which hath wisdom. The seven heads are seven
mountains, on which the woman sitteth. 10 And there are seven kings:
five are fallen, and one is, and the other is not yet come; and when
he cometh, he must continue a short space. 11 And the beast that was,
and is not, even he is the eighth, and is of the seven, and goeth into
perdition.
Since we have identified Rome as the "Head That Is"(the 6th Head), what are the previous 5? Furthermore, what is the 7th Head, which precedes the "Horns" that rule in Rev. 13:1?
Before I get into the identities of these Heads, there are 2 caveats which need mentioning: 1) the Beast Was and Is Not-therefore Rome, as much as we would like to make it to be, cannot be the Beast. Rome is certainly represented in the Beast, and is an important part of it's progression. But we cannot make Rome the "Beast" because Scripture doesn't make her the Beast; no matter how much we see Catholicism, the Pope, past injustices and present apostasy. Many use the "woman that sits on 7 Mountains"(Rev. 17:9) as the basis; using the 7 Roman Hills. Istanbul also has 7 Hills and so does Seattle, Washington. Furthermore, we cannot use "Literal" Mountains to describe a "Figurative" Beast, or worse yet, a "Literal" woman(Rome/Roman Catholicism) who's butt covers 7 Hills. Many make the mistake of using Literal and Figurative Interpretation in the same passage; but Context is the key. If we are talking "Literal" Mountains, then we are talking a Literal "woman" and a Literal Beast. If we are talking "Figuratively", and the passage "also are seven kings...."(kings=mountains) means the Context is Figurative, we cannot change from Figurative to Literal whenever it suits our interpretation. To summarize-the Beast, nor the woman is Rome.
The 2nd caveat is it is still a "Beast with 7 Heads and 10 Horns", and not a "Beast with 8 Heads and 10 Horns". Some use the "Revived Roman Empire" to qualify as the "8th Head", but that is not what the passage is saying. The "Beast" revives, but if we remember, "it is not..." during the Roman Empire. We can't 'revive' what was dead in the 1st place.
The Most Important Word in Dan. 7
To truly grasp the meaning of the Beast, understand it's nature, and remove speculation about individuals, systems, countries, etc. that attempt to cloud our thinking concerning this topic we must understand 1 singularly important word: that word is Dominion. The word "וְשָׁלְטָ֖ן"(wə·šā·lə·ṭān), which is a derivative of שָׁלְטָן(sholtan), comes from the root word שָׁלַט(shalat), which the BDB says in it's verb form to "domineer, be master of".
The word "dominion" is used 7 times in Dan. 7; it is 1st used in vs 6, when describing the 3rd Beast(Greek Empire), then it is used in vss 12 and 24, when the kingdoms' dominion is taken away, then in vss 14 and 27, when it describes the Lord's dominion, who after destroying the dominion of the beast, will reign forever, and all nations will serve and obey Him.
To understand "dominion", as it's only mentioned in the Greek Kingdom, is to contrast that understanding with the Lord's dominion. Both dominions have rule over the entire earth, but it's much more than land mass, physical territory or a particular ruler imposing his will. The Lord's dominion is the Kingdom of God, with it's laws, statutes, truths, ceremonies, feasts, as well as recognizing it is the Lord who exercises authority over every aspect of His domain. The Greek Kingdom, through Alexander the Great, did much more than conquer territory; it influenced culture, language, architecture, drama, literature, science, government, philosophy, theology, law, learning....all in celebrating the triumph of man, and the advancement of mankind. This model of thinking continues today, our scientific progress, governmental systems, ethics and legal systems all had their origins in the Greek Ideal. The fact that the 4th Beast(Rome), ascended to power and defeated the Greeks doesn't diminish this fact, they merely copied the Greek Ideal, and their pantheon, and gave it Latin names. There are 2 dominions being contrasted in Daniel 7: the Greek dominion and the Kingdom of God dominion. One dominion has the Christ as King, the other dominion, the Antichrist. One is the religion of man, the other the religion of God. The Greek dominion will ultimately be destroyed, and Christ's dominion will reign over the entire earth.
The Heads of Rev. 13
Understanding the word "dominion" greatly assists us in understanding Rev. 13. Rev. 13:2 says,
And the beast which I saw was like unto a leopard, and his feet were
as the feet of a bear, and his mouth as the mouth of a lion: and the
dragon gave him his power, and his seat, and great authority.
Since the "leopard" kingdom is Greece, the nature, characteristics, culture, learning, laws, science, government are Greek. Therefore the identity of this Beast is Greek-it is the 8th Head, yet 1 of the seven. Since the "lion"(1st beast of Daniel) is mentioned, it has it's roots in the Babylonian system, of which "soared to greatness" until it's wings were 'clipped', and most importantly, stood on it's feet like a man-representing the Kingdom of Man, as opposed to the Kingdom of God. It is no accident that it has the "feet of a bear", because the 2nd beast of Daniel is the bear(Medes/Persians). It leaned to 1 side, meaning the Persian kingdom gained dominance. But neither the "mouth" or "feet" are heads, meaning neither Babylon or Mede/Persia can be counted as heads, for they are already described.
Vision of the 3rd Beast
After this I beheld, and lo another, like a leopard, which had upon the back of it four wings of a fowl; the beast had also four heads; and dominion was given to it.(Dan. 7:6)
The 1st 5 Heads of the 7 Headed Beast are the Leopard(Alexander the Great-Greek) and the 4 Heads that followed his empire, which were divided among his 4 generals. The 6th Head(which is, during the time of John as previously discussed) was Rome-the 4th Beast of Daniel, from which the horns, which had no crowns in Rev. 12:3 as they were still colonies. The 7th Head, which in the statue of Nebuchadnezzar followed the legs of iron(Rome), was the Holy Roman Empire. The height of it's reign was the 3rd Reich, but it only lasted 13 years.
The 1st Head was "severed unto death", yet lived. Alexander the Great, and the "greatness" of the Greek Kingdom collapsed at his death, as there was no heir. The rival generals fought each other and lost the territory(and influence) gained during his conquest. Rome was a "terrible" beast, crushing all that didn't succumb to it's influence, and unlike the Greeks, held it's rule for 1500 years, although greatly diminished after the Fall of Rome.
But something happened, which 'revived' the 1st Head. That "revival" was called the Renaissance, and it has been recognized by most scholars as the beginning of the Modern Age. The "Greek" Head, diminished and depleted, now surged forth with renewed strength, invading the church, then the world. The scientific progress, educational philosophy, art, drama, poetry, along with law and government were all dramatically influenced by this movement. The Humanist Philosophers(Voltaire, Rousseau, Kant, Pope) eventually saw the "death of God" and sought to eradicate His influence in government, education, art and literature, and their persistence led to the revolutionary changes we see today.
It must be noted that each "Head" is not a person, but a "kingdom". Alexander the Great represented the Greek Kingdom, his followers represented the division of that kingdom in the 4 Cardinal directions(N,S,E,W).
Summary
Therefore, the "Head severed unto death, yet lived" is the Greek Kingdom. The Historicist View allows one to "see" through the progression of history, it's life, death, and resurrection into our present age. The interpretive clues, along with understanding "Dominion" show it to be diametrically opposed to the Kingdom of God.
*Note: Dispensational and Preterism also agree with this view. Futurism challenges it because it isn't "progressive"(Rev. 12 'should' be in the middle of tribulation period), although most Futurists agree the "man-child" is representative of Christ.(A further discussion was originally given, but limited due to exceeding format length.)