Yakuza. I've only played and beaten the first one. I remember renting when it came out with no expectations. Felt like a spiritual successor to Shenmue, but even better. Gameplay may be dated but I remember getting sucked in and couldn't stop playing.
As mentioned above, I think Shadow of the Colossus is a classic.
Jak and Daxter trilogy. I only beat the first one. Nice connected world platformer. The 2nd and 3rd are quite a departure. Add guns and open world.
Rise to Honor is a fun beatem up twin stick controls. I liked it a lot.
Simpsons Hit and Run. Some frustrating missions. But the writing and references are great if you're an old simpsons fan.
Psy Ops. I never got too deep into this game but loved the mind control mechanic.
And some quick nods: God of War, Prince of Persia, Mercenaries, Spiderman 2, Burnout 3, Gran Turismo 3/4, Viewtiful Joe, Dead or Alive 2, Soul Calibur 2, Tekken 4.