BIO HAZARD was the game that made me stay with the PSX. My first game was Street Fighter Zero (yeah I saw the BIO commercial in it) and it is also my favorite Street Fighter game (even with SF2 with countless upgrades still being popular in arcades).
Another game that was pretty good was Tomba! 2. One of the best 3D platformers IMO. My copy was NTSC-J, so I really didn't understand what was going on, but I got surprisingly far (to the palace level). For those who are not familiar with the game, it is a 2.5D world where you can do various quests and secrets. Due to the nature of the game and the Japanese language I would get obsessed with it, I remember getting real sad when I lost my copy. But I really didn't knew that there was an US version of the game and I wasn't able to play it even though I had a chip to play imports. One day I will track it down and finish it.
Today I only play some of my favorite ones on PS3: Resident Evil 1~3, Dino Crisis 1~2, Silent Hill, Alone in the Dark, Parasite Eve 2 (I'm not much of a fan of RPG games so I didn't got interested in the first one), all from PSN, and Alien Trilogy (disc). Also I would like to revisit Die Hard Trilogy, Need for Speed Hot Pursuit, and others that I can't recall right now.