Out of my considerably and alarmingly growing PS2 collection...
GTA San Andreas: A magazine gave it 98% percent when it came out... well it IS fun. I still play it to this day and enjoy it.
That being said, ANY GTA game is fun.
Burnout 2: Point of Impact: The Burnout I played most, doing over a hundred consecutive laps (!) in time challenge just to shave off a couple of 1/100 seconds... no other racing game has since hooked me like this one.
Tony Hawks Pro Skater 3 / Underground 1: 3 is better than 4, and Underground is cool for including three THPS2 levels.
Giants: Citizen Kabuto: Easily my most favorite PC game, and it does well on the PS2, too. The humor is awesome, the story original and funny, and its a very creative game.
God of War 1 and 2: Two pushed the PS2 beyond its physical capabilities, while being an awesome game down to the last pixel. I LOVE the God of War franchise, and while HD collections are out there, the PS2 titles STILL look and play fantastic. An absolute MUST.
Need for Speed Undetrground 1 / Most Wanted: Underground 1 was da bomb in '03, when tuning games where all the rage. Sadly the seven lap races are infamous for killing countless controllers in rage. But its (one of) the earliest tuning racing games thats decent - Underground 2 is rubbish.
Most Wanted on the other hand is fascinating. My most favorite is Need for Speed 3 Hot Pursuit, and Most Wanted comes dangerously close to its pole position. Almost.
The Simpsons: Hit & Run: A Simpsons GTA clone if you will, but brilliantly done! Has a few annoying fetch missions and tight as balls time limits (the last mission is plain evil but never to the point of frustration, you just want to try and try till you've done it). One of the few games I 100%ed.
Splinter Cell 1 / Pandora Tomorrow: Awesome stealth shooters, difficult, yet fun and engaging to play. Surprisingly well playable with controllers.
SSX Tricky / SSX3: Tricky was my first PS2 game I bought at the tender age of 14. I played the living **** out of it. I still fire it up these days, still knowing everything about every course, still being blown away by it. SSX3 just raises the bar like the badass game it is. I mean, snowboard down a mountain for an actual half hour? Shut up and take my money!