I' m bored, so here' s a few PS2 exclusives off the top of my head that are excellent:
Shadow of Destiny (mystery/adventure)
Onimusha 2, 3 (action; best action games available next to NG and DMC)
Devil May Cry (action; along with NG, a CANNOT-miss experience)
Twisted Metal: Black (action; 9.6 at GameSpot)
ICO (adventure)
Soul Reaver 2 (action)
Suikoden III (RPG)
Final Fantasy X (RPG)
Final Fantasy X-2 (RPG)
Ace Combat 4, 5 (flight)
Disgaea: Hour of Darkness, La Pucelle Tactics, Phantom Brave (Strategy/RPG)
Amplitude (rhythm)
Gran Turismo 3 (racing; still the best simulator out there, GT 4 coming in Dec.)
Ratchet and Clank, R&C: Going Commando (platform)
Jak and Daxter, Jak II (platform)
Kingdom Hearts (RPG)
Champions of Norrath (RPG)
Frequency (rhythm)
Gitaroo-Man (rhythm)
Gradius V (arcade/shooter)
The Bouncer (action)
Star Ocean: ' Til the End of Time (RPG)
Virtua Fighter 4: Evolution (fighting)
SOCOM II (action/strategy, online)
Zone of the Enders: The 2cd Runner (action)
Maximo: Army of Zin (action)
Guilty Gear X2 (fighting)
Rygar (action)
Silent Hill 3 (survival/horror)
Smugglers Run 1, 2 (action)
Super Bust-a-Move 2 (puzzle)
Tenchu: Wrath of Heaven (action/stealth)
Legaia 2: Duel Saga (RPG)