I’ll recommend Kinetica. It’s a one-of-a-kind racing game where you race through gravity-defying tracks as a person in a kind of iron-man negligee with wheels while listening to old-school techno.

Shadow of the Colossus is one of my favorite games ever, battling entities big enough that you run around on top of them, subtle storytelling, an enormous map for the time it was made, and fairly large even by modern standards.

The Tenchu games are also good: ninja stealth assassination.

Dark Cloud 2 is a kind of fun game. Smack your way through dungeons with a wrench and use the bits to build villages for your allies.

Bloody Roar is a favorite for fighting games. Fight to BIOS energy then transform into a wilder form, like a mole, a bear, etc. and you can kick people through the edges of the arenas into new areas to fight.

Devil May Cry is a classic.

Ratchet and Clank, classic.

Time Splitters is reminiscent of even older games.

Red Faction 2 wasn’t a bad little shooter.