this is like the ninth thread like this I've seen.
There are a LOT of good games for the PS2. If you are considering getting one, you could pretty much buy anything in the $9.99-$14.99 range and not be disappointed.
That said?
The Sly Cooper and Jak & Daxter series do a great job showing off how far the PS2 really brought videogaming. Each series' first installment is more or less Mario 64, with a bunch of collection and fetch quests. As they grew up, though, Sly got more into team-based spy/stealth and Jak & Daxter became a kind of sci-fi, alternate universe GTA. Just Rad.
If you were into JRPGs, you'd own the PS2 already. Because it is still effing heaven for those dudes. Persona, Final Fantasy, Disgaea, Rogue Galaxy? if you're into looking at overdesigned characters and waiting your turn , look no further.
It was/is also single player combat-heavy wonderland. Prince of Persia, Resident Evil 4 (Port, but whatever.) Devil May Cry (I didn't care about it at all, but tons of people did, so there you go.)
Did I just write 200 words after implying that there was nothing more to say on the subject? Yes.
If you have an extra sixty bucks and no current-gen games you want to buy, AND you don't care about multi player, AND you want to feel like your gaming collection is complete, AND/OR you want to take an academic view at your chosen form of leisure, you should not be without a PS2.
Or you could just find a backwards compatible PS3. but beware. Those things are like nerd bait on craigslist. I met so many robots trying to buy mine.