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.
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.
I think that game was probably my first real critique of capitalism. I played it at a young age and just fell in love with the anti-consumerist and liberation philosophy, though I didn’t have those words to describe it into much later in life. I really recommend the original Oddyssey and Exoddus over New m Tasty and Soulstorm. New n Tasty covers the same story points, but loses a lot of the environmental storytelling in the process, and Soulstorm is just a completely different game to Exoddus entirely.
It's still fun. Some parts are really difficult and feel somewhat unfair, but there's always a way to progress, so you shouldn't be discouraged when you encounter these points in the game.
Most of the things Psygnosis did; the Wipeout series, G-Police and Colony Wars being the most obvious choices. Tomb Raider is another good one, as are the Tony Hawk games. X-COM: Enemy Unknown is meant to be a solid port of the classic turn-based tactics game. I've heard praise for the Twisted Metal games and Syphon Filter, but haven't played them myself. While I know that old sports games are often considered anathema in the retrogaming sphere, Jonah Lomu Rugby is often considered one of the best games depicting rugby union ever made.
I had to think for a minute, I didn't realise how many of the console's stand-out titles are Japanese. Crash Bandicoot: Warped, Tony Hawk's Pro Skater 2, and Driver are my picks.
I've heard the PS1 port of Doom is good/interesting. If you haven't played it before (or even if you have) it might be worth checking out. It has different lighting and textures, from my understanding.
Panzer Front. Tank game set during WWII that also had strategy elements, such as setting routes for the AI tanks and updating new ones. Pretty hard, but a really great, tense atmosphere.
Unless you mean exclusives, I think if you can get over the initial clunkyness the original Tomb Raider games are some of the best action adventure games ever made at the time. Her movement starts to feel really good once you get into it.