Does Gandalf literally know he's a type of angelic spirit called a Maia, who is a servant of a greater Vala called Manwë, who was given a corporeal form and sent to Middle Earth?
Or did he appear in Middle Earth as an old wizard with only vague impressions of where he came from, rather than explicit knowledge?
What evidence is there in the text?
Almost certainly. In the Two Towers (The Window on the West), Faramir quotes Gandalf as saying
Many are my names in many countries ... Mithrandir among the Elves; Tharkun to the Dwarves; Olorin I was in my youth in the West that is forgotten, in the South Incanus; in the North Gandalf; to the East I go not.
A similar quote appears in The Quest of Erebor (Unfinished Tales), where Gandalf says
But what I knew in my heart, or knew before I stepped on these grey shores: that is another matter. Olorin I was in the West that is forgotten, and only to those who are there shall I speak more openly.
Also in the Unfinished Tales (the Istari) we find
... though they [the Istari] knew whence they came, the memory of the Blessed Realm was to them a vision from afar off ...
From these it's clear that being embodied as an Istar may have caused Gandalf to forget some things. However, he certainly knows where he comes from, and from this he could guess his true nature, or Galadriel or Elrond could tell him. Indeed, in Of the Rings of Power and the Third Age, Galadriel favours Gandalf as head of the White Council (indicating that he had the greatest wisdom in a group that included both Galadriel and Elrond), but he refuses
since he would have no ties and no allegiance save to those who sent him.
Gandalf also knew that he carried considerable power, since he was prepared to fight a balrog, which was too great a foe even for Aragorn. He told the balrog
I am a servant of the Secret Fire, wielder of the flame of Anor.
I don't think these exact phrases are ever used elsewhere, but they probably refer to the Imperishable Flame (see the Ainulindale).
What else could Gandalf believe himself to be?
