What is the "verbal plenary" view of the inspiration of Scripture and to what texts is it considered to apply? What hermeneutical approaches hold this view and how does it affect the way they prioritize different manuscripts in their pursuit of the meaning of a given passage?
Verbal Plenary Inspiration means that the very words of the original manuscripts were directly inspired by God, although he did not change the author's intelligence or understanding while doing so. Meaning, that God choose, inspired and sovereignly guided the biblical authors who were equipped by Him to write the very words of scripture. This is the foundation of the evangelical view of the inerrancy of scripture.
This should only apply to the canonical books within the tradition in which the doctrine is accepted. As a Protestant I believe this applies only to the canonical books of the Protestant Bible.
This is a view mainly held by Protestants, so I would offer this as a general statement: if you hear someone espouse this view then they are probably referring to the Protestant Bible. If I am incorrect I would gladly take correction, but I do not think my Catholic and Eastern Orthodox brothers would agree with this doctrine or apply it to their canon.
From Wikipedia as an outside source:
Evangelicals see the Bible as a truly human product whose creation was superintended by the Holy Spirit, preserving the authors' works from error without eliminating their specific concerns, situation, or style. This divine involvement, they say, allowed the biblical writer to reveal God's own message to the immediate recipients of the writings and to those who would come later, communicating God's message without corrupting it. Some Evangelicals have sought to characterize the conservative or traditional view as verbal, plenary inspiration in the original manuscripts, by which they mean that every word (not just the overarching ideas or concepts) is meaningfully chosen under the superintendence of God. Evangelicals acknowledge that there is textual variation between accounts of apparently identical events and speeches, which would seem to have God saying different things. Some of these differences are accounted for as deviations from the autographa that were introduced by copyists, while other cases are considered intentional deviations that were inspired by God for particular purposes (for instance, the Gospel of Matthew was intended to communicate the Gospel to Jews, while the Gospel of Luke was intended to communicate it to non-Jews). Many Evangelicals consider biblical inerrancy and/or biblical infallibility to be the necessary consequence of the Bible's doctrine of inspiration (see, for example, the Westminster Confession of Faith or the Chicago Statement on Biblical Inerrancy), though not all do.
Note: Although I think there is some evident bias against this doctrine in the Wikipedia entry it does represent a valid, secular look at the idea.
We all know the meaning of the word "verbal", but there may by different understandings of the word "plenary." The root/origin of the word "plenary," from late latin (plenus, plenarius) and into the middle ages when the word "plenary" was forged, had much to do with the concepts of "fullness" or "completeness/comprehensiveness." So the meaning of the entire phrase, "verbal plenary inspiration" would reference the actual words (verbal) and their comprehensive function for salvation and human living (plenary) coming directly from the spirit of God to the writer (inspiration). If you are referencing mainstream protestantism where the phrase in your question is used only to refer to the fully canonized books of the Bible, then that "plenary inspiration", or "comprehensive guidance of the spirit of God" must be extended to the councils that assembled the collection of books into what we reference today as "the Bible".
However, there are Christians who claim that prophecy is an active gift of the church through all time, and that, as stated in the book of Joel and quoted by Paul, "in the last days your young men shall see visions and your old men shall dream dreams" (Joel 2:28 and Acts 2:17). Also, the book of Daniel speaks of "sealing up the books (of prophecy) until the end of time" (Daniel 12:4) and a corresponding command found later in Revelation, "do not seal up the words of this prophecy for the time is near" (Revelation 22:10) imply that there is perhaps, at least "understandings" or "interpretations" of what is found there that are yet to be revealed. This might present a conundrum for those who insist on the "verbal plenary view of inspiration" and also insist on the veracity of Joel/Paul's words concerning new prophetic words to be given toward the end of days.