I do like myself a good Japanese game every once in a while, but right now I'm looking for other PS1 games made outside of Japan that are still very fun to play.
There's a lot of great Western developed games on the PSX. I think on average they did a better job than on the consoles a generation before. The Tomb Raider games are probably the standout series, the first one is one of the best games on the system. Another big series I didn't see mentioned is GTA. Maybe the old games are more associated with PC? I played them on PSX and really enjoyed them despite all the slowdown.
Heart of Darkness is a nice platformer I just remembered. Also, V-Rally 2 was a pretty good racing game.
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.