1) FPS - Goldeneye
2) Third Person Shooter - Gears of War
3) Adventure - Psychonauts
4) RPG - The World Ends With You
5) RTS - Lord of the Rings: Battle for Middle Earth
6) Racing - Diddy Kong Racing
7) Sports - WWE Smackdown: Here Comes The Pain
8) Online - World of Warcraft
9) Puzzle - Scribblenauts... if it counts.
Can't help but notice you forgot Fighting...
Fighting added.
My favourite games from that genre are Mortal Kombat 4 and each and every WWF/E game.