Aww man. This is tricky. I'll cheat and do it by console
PC: Lego Racers, though Star Wars: Galactic Battlegrounds gets an honourable mention for introducing me to the RTS genre.
PS1: Rainbow Six 1
SNES: (Street Fighter) Final Fight II
Game Boy: Doraemon 2
I remember playing Rayman 2 but I eventually stopped playing cos I got stuck :/ It's crazy that platform/puzzle games were big titles a decade or so ago, but now it'd unheard of to have a blockbuster platformer without being dressing up with some kind of FPS/action element.
Okay, Mirror's Edge and Portal come pretty close; and the Mario franchise has always been successful, but the successful platformers nowadays are mostly (underated) indie titles. Ohh the nostalgia...