The word 'archangel' (lemma: ἀρχάγγελος) appears twice in the New Testament, at least once in the LXX translation of the book of Enoch (which mentions numerous angels and their duties and authority - I would read it here or here if this topic interests you), and also in a highly disputed verse in 2 Esdras. It should be noted that after the fourth century, few Christians considered Enoch to be canonical.
- 2 Esdras 4:36 - "And unto these things Uriel[*] the archangel gave them answer, and said, Even when the number of seeds is filled in you: for he hath weighed the world in the balance." *The RSV renders the archangel's name as Jeremiel (an alternate transliteration). It should be kept in mind that this is a highly disputed text (both the KJV and RSV w/apocrypha call this book '2 Esdras,' some scholars and other traditions call it 4 Esdras or 'Latin Esdras,' as the only copy preserved is in Latin).
- Enoch 20:7 - All of the main Greek texts conclude this verse with ἀρχαγγέλων ὀνόματα ἑπτά. However, no English translation that I've found translates this phrase, which could be translated, "Of archangels there are seven names," or potentially also "There are seven names of archangels." See Swete's 1909 LXX critical text and apparatus for this verse. There may actually be seven archangels (although I will not assert this as it is purely speculation). Various church traditions include up to 14 archangels.
- Enoch 9:1, 4 - The word 'archangel' only appears in what Swete labels as alternate texts - it is not in the primary texts that he chose for his critical edition of the LXX. I still wanted to mention this because the alternate texts would potentially be labeling "Michael, Gabriel, Raphael, and Uriel" as archangels, which would mean that there is more than one (other manuscripts appear to also include Suryal). See Swete's 1909 LXX critical text and apparatus for these verses here and here.
- 1 Thessalonians 4:16 - "For the Lord himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God...." There is no definite article prefacing 'archangel' in the Greek, so this text should not be construed to imply that there is only one archangel.
- Jude 9 - "Yet Michael the archangel, when contending with the devil he disputed about the body of Moses, durst not bring against him a railing accusation, but said, The Lord rebuke thee." It should be noted that Jude 14-15 is a quotation from Enoch 1:9 (a non-related section), and so it is significant to use evidence in Enoch since Jude quotes from the book. Jude 9 may also have been alluding to Enoch 9 since the angels (including Michael) did not directly accuse Azazel and Semjaza, but rather went to the Lord. However, most Christian scholars ascribe this to a questionable work entitled The Assumption of Moses (or Testament of Moses). With any discussion of Jude, we should also examine possible "parallel" verses in 2 Peter (for this reason). 2 Peter 2:11 says, "Whereas angels, which are greater in power and might, bring not railing accusation against them before the Lord." This is also likely an allusion to Enoch 9. Jude and Peter are traditionally said to be discussing the punishment for the angels responsible for the Nephilim in Genesis 6 (historically thought to be an instance of incubus).
Disclaimer / Limitation: I am working with a critical text of the book of Enoch from 1909 (Swete). I do not have any critical texts for Enoch that are more recent. I found the Skeptik Greek text for Enoch online when researching this which supports my finding for Enoch 20:7. However, it seems that many English translations had to have been working from different manuscripts, as there is an additional name in 9:1 and other phrases throughout Enoch that were not in the Greek manuscripts that I had available (unless these translations are faulty, which is also a possibility - but I have not conducted enough research to make this assertion). But the main translation that differs from Swete's Greek text (by R.H. Charles) was published in 1913, so theoretically the same manuscripts should have been available. It should also be noted that I did not have a Greek manuscript for the verse in 2 Esdras (there may not be one), I used the English KJV translation.
Meaning of the Word
The ἀρχ- prefix is associated with the Greek word ἄρχων, meaning "ruler" (a substantival participle literally meaning "ruling one"). It is used in interesting ways throughout scripture and early Christian history in reference to angels. The word derives from ἀρχή, which is usually translated as "beginning" (which is the gloss), but can also mean,
"an authority figure who initiates activity or process, ruler,
authority.... Also of angelic or transcendent powers, since they were
thought of as having a political organization (Damascius, Princ. 96
R.) Ro 8:38; 1 Cor 15:24; Eph 1:21; 3:10; 6:12; Col 1:16; 2:10,
(such as its peculiar use in Jude 6 which is relevant to this discussion).
The word also appears in other early Greek literature in case you're interested in the references:
ἀρχάγγελος, ου, ὁ (s. ἀρχή, ἄγγελος; En 20:8; TestSol; TestAbr A B;
TestLevi 3:5 v.l.; ParJer 9:5; GrBar; ApcEsdr 1:3 p. 24, 7 Tdf.;
ApcSed 14:1 p. 135, 33 Ja.; ApcMos; AssMos Fgm. k; Philo, Confus.
Lingu. 146, Rer. Div. Her. 205, Somn. 1, 157; Porphyr., Ep. ad
Anebonem [GParthey, Iambl. De Myst. Lib. 1857 p. xxix–xlv] c. 10; cp.
Iambl., Myst. 2, 3 p. 70, 10; Theologumena Arithmetica ed. VdeFalco
1922, p. 57, 7; Agathias: Anth. Pal. 1, 36, 1; ins in CB I/2 557 no.
434 ὁ θεὸς τῶν ἀρχαγγέλων; Gnost. ins, CIG 2895; PGM 1, 208; 3, 339;
4, 1203; 2357; 3052; 7, 257 τῷ κυρίῳ μου τῷ ἀρχαγγέλῳ Μιχαήλ; 13, 257;
328; 744) a member of the higher ranks in the celestial hierarchy,
chief angel, archangel PtK 2 p. 14, 27. Michael (En 20:5; 8; ParJer
9:5) is one of them Jd 9. He is also prob. the archangel who will
appear at the last judgment 1 Th 4:16 (the anonymous sing. as PGM 4,
483, where the archangel appears as a helper of Helios Mithras).—See
WLueken, D. Erzengel Michael 1898; Rtzst., Mysterienrel.3 171, 2;
UHolzmeister, Verb. Dom. 23, ’43, 176–86; s. on ἄγγελος.—149–53. M-M.
Jesus = Michael?
Concerning the equation of Jesus with Michael, some scholars3 believe that the early Christian book The Shepherd of Hermas (which was considered canonical by many Christians in the second and third centuries, although not without much controversy) equates Jesus with Michael (this is disputed among historians)4. Primitive Christology understood Christ as the "prince of angels," but this debate was refuted by later debates in Christian history and The Shepherd of Hermas was rejected as heretical (both for teaching Adoptionism and for equating Jesus with Michael), especially on the basis of Hebrews 1 which had much greater support in post-Nicene Christianity.5 The equation of Jesus with Michael did not become a topic of contention in Christian history again until the 19th century.
The answer to your question really depends on what books you consider to be the Bible. Assuming you are operating with a mainstream historic Christian understanding of the biblical canon6 (which would exclude Enoch), the Bible does not specifically name any other archangels other than Michael. However, on the basis of other Christian texts such as Enoch as well as Jewish and Christian history, it would not be advisable to consider Michael to be the only archangel. The Biblical text does not imply this anywhere (it makes no implications one way or the other), nor is it supported by other texts and historic traditions.
1 William Arndt, Frederick W. Danker and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 3rd ed. (Chicago: University of Chicago Press, 2000), 138.
2 Ibid., 137.
3 Jaroslav Pelikan, The Christian Tradition: A History of the Development of Doctrine: Volume One - The Emergence of the Catholic Tradition (100-600), (Chicago: University of Chicago Press, 1971), 183.
4 cf. Shepherd of Hermas 69:3. Relevant quote produced below:
"And the great and glorious angel Michael is he who has authority over this people, and governs them; for this is he who gave them the law into the hearts of believers: he accordingly superintends them to whom he gave it, to see if they have kept the same."
"The Pastor of Hermas", trans. F. Crombie, in The Ante-Nicene Fathers, Volume II: Fathers of the Second Century: Hermas, Tatian, Athenagoras, Theophilus, and Clement of Alexandria (Entire), ed. Alexander Roberts, James Donaldson and A. Cleveland Coxe (Buffalo, NY: Christian Literature Company, 1885), 40. This can be read online.
5 Pelikan, 183-184.
6 It is regarded as canonical by the Ethiopian Orthodox Tewahedo Church and the Eritrean Orthodox Tewahedo Church. The Church of Jesus Christ of Latter-day Saints (LDS/Mormons) does not consider it to be canonical but does consider it to be inspired. Everyone else excludes it (including most of the Eastern Orthodox Church except for those noted above).