Iluvitar made 2 classes of spirit beings to sing the songs of creation. The higher class (more powerful) are the Valar who are 15 in number, the lower class (less powerful) are the Maiar. I can find the names of about 15 or so Maia but the stories seem to indicate there are many more than just 15 of them. How many Maiar did Iluvitar make? Dozens? Hundreds? Thousands?
Not known; Tolkien himself wrote that the Elves didn't know how many (emphasis mine):
With the Valar came other spirits whose being also began before the World, of the same order as the Valar but of less degree. These are the Maiar, the people of the Valar, and their servants and helpers. Their number is not known to the Elves, and few have names in any of the tongues of the Children of Ilúvatar; for though it is otherwise in Aman, in Middle-earth the Maiar have seldom appeared in form visible to Elves and Men.
The Silmarillion II Valaquenta "Of the Maiar"
I don't know of any writings where Tolkien gives an estimate, so this is what we have.
A bunch; at least fourteen or fifteen named:
- The five Istari
- Eönwë, herald of Manwë
- Ilmarë, handmaid of Varda
- Ossë, vassal of Ulmo and Uinen, his wife
- Salmar, of Ulmo
- Arien, the Sun
- Tilion, the Moon
- The balrogs, of whom there were either seven or "a lot"
- Probably Huan
- Some number who became orcs, some of whom may or may not have been called Boldog
And possibly any number of other creatures; it's a fairly common assertion among Tolkien fans that any speaking creature that isn't a Vala, Man (or Hobbit), Elf, Dwarf, or Orcish is necessarily a Maia; that would include (but is not necessarily limited to):