GTA San Andreas, Liberty City Stories and Vice City Stories
God of War 1 and 2
Devil May Cry 1 and 3
Max Payne 1 and 2
Ratchet and Clank
Jak and Daxter 1, 2 and 3
Shadow of the Colossus

Legacy of Kain: Blood Omen, Soul Reaver: Legacy of Kain, Soul Reaver 2: Legacy of Kain, Legacy of Kain: Blood Omen 2, and Legacy of Kain Defiance

In that order, of course.

If you like traditional JRPGS, go with Dragon Quest VIII. It's a classic of its era.

THIS!!!

It's the only game you need on PS2 if you pick just ONE!

Dragon Quest VIII (8) !!!! The game is sooo amazing (though its getting a 3DS version with some additional content)
Kingdom Hearts 2 (though it has a ps3 remaster)
Kingdom Hearts (also has a ps3 remaster)
Shadow of the Colossus - Unique and amazing.
Final Fantasy XII
Final Fantasy X
Tales of the Abyss
Ico
Jak and Daxter: The Precursor Legacy
Devil May Cry 1 and 3´
Ratchet and Clank games, especially Up your Arsenal
Sly Cooper 1 and 2 (I don't know how good is 3)
God of War 1 and 2
Dark Cloud 2

PS2 is still my favorite console

Yep. VIII is great. Battles start and end quickly. Excellent pacing that breaks the random battles up, between towns, and exploring. Tons of hidden secrets around every corner. VIII is almost as good as Chrono Trigger, and FFVI.