Pretty much the only reason I bought a PS1 was for:
Ridge Racer! I fell in Love with that game in the Arcade.. and the PS1 port is excellent in playabillity, difficulty/challenge, and sound. Clear all the Invaders in the beginning, and unlock more cars. Beat the full game, and you then get to race everything in the opposite direction. Beat the black #13 Car, win the game.. and it unlocks it for use. The 13 car will actually let you lap it once or twice, before it gets going. Its so fast, that it can easily still win the race... unless you are mad skilled at keeping him behind your bumper. One skip up on a turn... and hes gone for good.
I think I had also picked up:
Warhawk (meh)
Tomb Raider (horrible camera issues = bad experience)
MK Trilogy? (Load times killed the fun)
Philosoma ? An interesting Shump that had many various perspectives. Fairly cool.. but I recall it being a bear... and its one of the few games that I have never beaten. I always swear Id fire it up again and try once more... but I never seem to have the time. I think there may have been some cheap-ness about the game that ticked me off... and destroyed my will to play it again.
Rage Racer - Awesome game! A slow start, but challenging and deep. This game eventually gets Real fast, and real intense. Its still a blast to play.
Adventures of Lomax in Lemmingland - One of the few traditional style platformers. Gorgeous graphics that scale in and out of the screen. This game gets quite hard... and despite its child like looks... I think I recall it being quite a bear to beat. (though, I think I initially didnt use the powerfups well when I started out).
I really didnt get into much more than that... as most of the games were 3d based, and didnt visually impress me. Im more of a 2d guy... except for racing games. (and nothing really beat the speed, fun, and challenge of RR and Rage Racer, IMO. I never could get into those SIM games.. as they just feel way too slow, and I didnt have a FFB wheel)
Its sad that Sony basically forced developers to produce all 3d games. Only way Lomax got around this, was to paste the 2d graphics on a 3d framework. That game itself is kinda rare I believe... and I dont recall any other similar games for the ps1.
Edit:
How could I forget?!
Rayman!
Another great platformer. Very tough. Unique. Graphically Luscious. Atmospheric. I wish they would have kept that same formula for the sequels.. but nothing since has captured that same difficulty, unique playability, and beautiful looking graphics.