This article is from the Tolkien Newsgroups FAQ, by Steuard Jensen sbjensen@midway.uchicago.edu with numerous contributions by others.
The answer depends on exactly what the question means. Below are
listed a number of possible answers (as of the end of the Third Age),
starting from the oldest.
1. Eru Iluvatar, the Creator... but he never inhabited Ea itself.
2. The Ainur (including Sauron, Gandalf, etc.): they existed before
the Music that gave Middle-earth form.
3. Tom Bombadil. In addition to his direct claim that he is
"Eldest" (confirmed at the Council of Elrond), he says that he
"was here before the river and the trees", and that he
"remembers the first raindrop and the first acorn". If he is
one of the Ainur, this implies that he was the first of them to
enter Middle-earth; if not, it probably means he was the first
"native" inhabitant.
4. Some trees in Fangorn (and maybe elsewhere): Treebeard says that
in some parts of his forest, "the trees are older than I am."
5. Treebeard. Gandalf tells Theoden that he is "the eldest and
chief of the Ents, and when you speak with him you will hear the
speech of the oldest of all living things." (Given #4, Gandalf
must actually mean something like "speaking living things", and
given #2 and #3 he must be using a specific definition of
"living".)
If any of the Fathers of the Dwarves were alive (having been
"reincarnated"), they might fall between #4 and #5. As any living Elf
would certainly be one of Gandalf's "living things", all of them must
be younger than Treebeard. (Although the Ents awoke only after the
Elves, this does not prove that none of the "First Elves" remained
alive: Treebeard could conceivably have existed as a normal tree before
awakening as an Ent.)
 
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