Damn, the best PS2-game ever? Hell, I wouldn't even be able to choose the best of a genre on that platform. The PS2 stood strong in pretty much every genre, but it was especially note-worthy when it came to platformers and JRPGs.
My list of candidates:
Shin Megami Tensei: Lucifer's Call (or SMT III: Nocturne, for you US-people)
Of all the SMT-titles (including Digital Devil, Devil summoner and Persona) this one still takes the cake. The stoic feel and use of the clair-obscure technique create an amazing atmosphere. It has a distinct occult feel that oozes throughout character design and storytelling. Plus, it's really lengthy, you can persuade over 180 demons and it happens to be quite challenging.
Shadow Hearts Covenant
An amazing JRPG that did everything its predecessor didn't. The original Shadow Hearts had a legendary soundtrack, a nice gameplay concept (the Judgement Ring) and excelled in both dialogue and story. However, the graphics were terrible and apart from the Judgement Ring (and the fusions), the gameplay was far from inventive. Enter: Shadow Hearts Covenant. The plot, the music, the graphical design, the combat... Damn, the combat: combo's, fusions, crest-based magic, the Key of Solomon, adaptable Judgement Rings and individual-bound moves. Oh, and while it didn't have the most detailed CG-movies, they did have the most cinematographically awesome ones, I kid you not!
Jak II: Renegade
A sequel that surpassed the original in every possible way. A distopian city made for free-roaming, a bunch of external areas and a great balance between combat (armed + melee) and platform elements. Not to mention the story, the cut-scenes and the humor. Oh God, the humor...
Ratchet & Clank 2 (or Ratchet & Clank: Going Commando, for you US-people)
Again, the second one in a series - seems like a conspiracy, don't you think? It would seem that adding some RPG-elements to Ratchet & Clank's original gun-fest was a REALLY good decision.
TimeSplitters 2 or TimeSplitters: Future Perfect (not console exclusive)
A short but powerful solo campaign, with Challenges and an Arcade mode to expand the single player experience. A surreal multiplayer (with 4-player offline option) that never got old: tons of weapons from various time periods, lots of variation in the maps, 13 different multiplayer modes, lots and lots of characters and the freedom to adapt almost everything in a match. It even had a map-maker, with the option to make your own missions. Gentlemen, we NEED a TimeSplitters 4.
Okami (for a while, it was exclusive)
Engaging, funny, inventive and a true marvel on the audio-visual department. The Celestial Brush was one of the most incredible ideas I ever experienced in gameplay.
Team ICO titles (ICO and Shadow Of The Colossus)
Just... Beautiful, nothing more to say about these titles, really. It's an experience.
And I suppose Final Fantasy X, Persona 3, God of War and Sly Raccoon deserve an honorable mention.