Oddworld: Abe's Oddysee and the sequel Abe's Exoddus. Easily one/two of the best games I've ever played. They're available on PC and other systems these days, but they're still the definitive Playstation game for me. Unique and one of a kind and even today, absolutely great to play.
I also give a vote to Legacy of Kain: Soul Reaver and Medievil. Surprised no-one's mentioned Oddworld though.
Edit: Ooh I have to add, Rayman! And if you have a friend to play with - Team Buddies.