I can't really pick one favorite PS1 game overall, but I guess I can for each genre. Also these picks are based on the games I played/owned when the PS1 was still alive. (Back then my PS1 library only consisted of 20 games.. today I own 328 PS1 games and my picks would probably be different.)
PS1 Horror - Resident Evil This was the first video game that ever made me actually jump. I remember renting this game from a Blockbuster multiple times before my parents ended up buying it for me on my birthday. I was so stoked because it was the long box version and I still have it today. Beat that game multiple times to see all the endings and also to get the rocket launcher with infinite ammo and the dressing room key... Plus I don't think any other game got me so anxious waiting for the sequel to come out as much as this game did. I would pick up every video game magazine at the time just to see if there were any screenshots or updates.
PS1 Sports - Thrasher: Skate and Destroy In high school I was really into skateboarding and some hip-hop so I fell in love with this game. Most of my friends chose Tony Hawks Pro Skater but to this day I still think this game was ahead of it's time. I couldn't stand how unrealistic the Tony Hawk game was.. you go off ramp, button mash and do like 50 kickflips.. Thrasher on the other hand was legit. You press one button to perform one realistic kickflip and (if you played on professional mode) you had to press another button to actually catch the board with your feet and brace for impact/landing. Also you got to choose your sponsors which I don't think came along in Tony Hawk until 2 or 3..
PS1 Driving - Destruction Derby 2 Playing this game with friends was a blast.. the arena course where all the cars drive down to the bottom and launch off a ramp and smash into each other in the air. I also loved how you could go first person when driving and how your screen would flip along with the car when rolling, I never really got tired of this game ..but that announcer though.. "Hey, go easy on the paint work!", "You've broken the rear axle!" .. "It's all over!"
PS1 Fighting - Tekken 2 I remember being able to beat all the kids on my block, especially if I played as Law. I also really enjoyed trying to beat the game with every character just so I could see those cheesy cg endings. But before I got this game I used to play Battle Arena Toshinden a lot, but unfortunately when juxtaposed it just doesn't hold up. Except for the music, I'll give it that.
PS1 Adventure - Mega Man Legends One of the few PS1 games I enjoyed so much that I beat the game in only 3 days. I also found all the pieces of his suit and completed all the side missions to get the ending where everyone is waving goodbye. I hoped that they would make a sequel, which they did but sadly I never got to play it. I think by that time I was already moving on to the PS2 or N64..