Shadow of the Colossus was the first game that gave me feels that I couldn't articulate. It re-framed my idea of what a game could be, and gently pushed me to think about games as a medium.
I was too young to play it when it came out, but watching my dad play it was the most mind blowing things to me as a kid. I was completely immersed in it then, and its still my favorite game of all time.
Dragon Ball Z Budokai Tenkaichi 3, this game is perfect as a Dragon Ball Z simulator, even for today's standards, I wanted to mention an exclusive title, but many of the PS2 games have already been ported to other platforms, which speaks very well about the PS2 itself.