Its awesome, but if you like racing games, Grand Turismo, is alright
Gran Turismo is absolutely seminal, probably the most influential racing game ever. It is, however, soundly thrashed by its sequels.
I don't want to rehash the same games everyone else is saying, so I'll suggest some others. Although I will point out that every main series Final Fantasy up to IX other than IIIj can be had on PS1.
Grand Theft Auto and
Grand Theft Auto 2. How the series started out, in the top-town 2D days. GTA2 is a lot more free-roaming than the newer games. These can, admittedly, be had for PC also.
The
Driver series can also be seen as a GTA antecedent. The games are all about car chases, mostly in oldish American muscle cars - there's no guns or shooting and in Driver 1 you can't even leave your car, so you get to focus on the driving. (Strict police too - no ramming or jumping red lights here!) Along a similar vein is
The Italian Job, a PS1 video game based on the film - the
original British film, not the LA heist remake.
The
Wipeout series is worth a look, for futuristic racing that's better than F-Zero.
A little-known gem is
Rollcage. A futuristic racer with a sci-fi setting like Wipeout, but the gameplay's been said to be more like Mario Kart.
Jade Cocoon is interesting. It's a bit of a Pokemon ripoff tbh, and I'd call it average-to-good, not brilliant. The plot is a bit meh, and the pace slow. The key thing that makes it not Pokemon is the ability to merge monsters, creating hybrids - and that being a key way to get a bigger range of abilities.
Another also-on-PC (and on the DS!) case is the
Worms series. I'd steer clear of the original, it's very dated. Worms Armageddon and Worms World Party are great.
For something not really a "game" as such there's the
Music series. Of course you could criticise them for doing what's better done on PC.
Lastly, a bunch of innovative puzzlers:
Kurushi AKA
Intelligent Qube,
Mr Domino, and
Devil Dice AKA Xi.