I actually wanted to post a PS2 appreciation thread, so I was glad to see this post pop up!
PS2 was fantastic, I preferred Gamecube as a console but PS2 had the games.... just an incredible library of unmissable GOATS.
Specific games that I played endlessly, were MGS 2 and 3, GT3, Tekken 4 and 5.
Of course I had a big enough library of games but those were games I kept coming back to.
I got the console with Tekken Tag and couldn't save till I got a memory card....so I often left the console on for a few days in a row.