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.

The Syphon Filter series

Tons of non Japan created PS1 games.

Crash Bandicoot was made by Naughty Dog software, a US based company.

Legacy of Kain and Soul Reaver were Canadian games.

Spyro the Dragon was from Insomniac games in California. Also Spiderman.

Tony Hawk Pro Skater was by Neversoft, again California.

Spyro, crash bandicoot and CTR, tomb raider.

Come to think of it, most of my played PS1 games come from Squaresoft somehow

Wipeout 2097 (Wipeout 1 had broken collision detection, no wall grinds. Avoid.)

Driver or Driver 2

Micro Machines (multiplayer best)

Twisted Metal 2 World Tour (multiplayer)

Destruction Derby

G-Police was not bad too.

Japanese recommendations : PaRappa or Tekken 3.

Avoid any FPS or FIFA, all laced with frustration.

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.

"Tunnel B1' , created in Germany. A very good game that many slept on...the difficulty in the later stages is ruthless though!

ODDWORLD! ODDWORLD!

Diablo

That's the only western game in my PS1 library.

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.

Twisted Metal 2 is my personal favorite

Team buddies

Have you played the MediEvil games?

Soul Reaver. The Colony Wars trilogy.

Wipeout series and Fighting Force (the first one).