I am on the hunt for Silent Hill 2 - one of the best games ever made, get the Directors Cut for PS2 if you can, it's got extra content.
My other Favourites and yet to start/beat are:
Aiming for 100% completion on these:
GTA III
GTA Vice City
GTA San Andreas
GTA LCS
GTA VCS
Story completion on these:
Timesplitters Future Perfect
Killzone
XIII
Red Faction 2
Need For Speed Underground 1
Need For Speed Underground 2 (slow load times but worth it)
Half-Life
Max Payne
Black
Splinter Cell
Splinter Cell Pandora Tomorrow
Splinter Cell Chaos Theory
Medal Of Honor Frontline
Medal Of Honor Rising Sun (slow load times but great game)
I am trying to spend less time on the web so playing an offline console like PS2 does me good.
GTA3 (Yes, I had more fun with this one than Vice City and San Andreas)
Tony Hawk Pro Skater 3
Shadow of the Colossus
Resident Evil 4
Silent Hill 2
Burnout 3: Takedown
Shadow of the Colossus is one of my favorite games from that generation, and my favorite from the PS2. (Apparently the PS4 port is worth a look, according to DigitalFoundry).
Okami is a close second.
I never had a PS2, but i picked them up for the PS3.