Wasn't on the list, so I decided to suggest it.
In it, you play as a young psychic named Rasputin (or Raz) in a camp for psychics called Whispering Rock. It's made by Tim Schafer, so you can expect good stuff from him. You can learn powers such as pyrokinesis (burn things with your mind), psychokinesis (lift things with your mind), and confusion (self explanatory). You can also meet interesting characters such as former secret agent Sasha Nien, party-loving Milla Vodello, and schizophrenic Ford Cruller.
You can play it on Xbox, OR you can get it on Steam, which includes new achievements and developer commentary, but you probably don't care about that.
Video (apologies for low quality):
youtube.com/watch?v=y-0yBX2wncM
Steam Link: store.steampowered.com/app/3830/
Screenshots (contains minor spoilers):

Alright, I've added some screenshots, as well as a more detailed explanation, but not too detailed; the fun of games by Tim Schafer is the sense of discovery.