I came up with a few examples, though perhaps fewer than I expected:
– Into the Wild, by Sarah Beth Durst (The heroine is Rapunzel’s daughter, but all that gives her is more knowledge than most of how fairy tales work, and no magic powers whatsoever.)
– The Homeward Bounders, by Diana Wynne Jones (the more I think about it, the more this one is a really good example)
– Enchantress from the Stars, by Sylvia Engdahl
– Siberia, by Ann Halam
– And Tolkien’s Lord of the Rings, of course. One of the first, best examples of a non-special, non-chosen character saving the world.
Other examples that you all can think of?