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.
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.
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.