Games

The Simpsons: hit & run, Shinobido, GoW, Destroy all humans.

I spent a ton of time playing the two towers, bit I don't know if it has aged well.

Also, war of the monsters was campy and fun.

Not exactly a suggestion, but why emulate Okami instead of just running the PC version?

Edit: If you get the PC version, you should be able to use the touch screen to draw the brush techniques directly thanks to mouse support rather than having to fuck with the sticks. You also don't get the input lag from emulation.

Id look up the GT4 spec II mod. it adds a bunch of features to the game and even the Valencia track from Tourist Trophy

Shining Force NEO and Shining Force EXA are both great underrated action RPGs.

Klonoa 2

Super Monkey Ball Deluxe

Beatmania IIDX (if you don't mind connecting a keyboard to play)

I see Rule of Rose, I upvote.

Road Trip Adventure

Fatal Frame II

Mister Mosquito

Gitaroo Man

It's a little cringe in retrospect, but I really liked my time with Makai Kingdom. Avoid if anime stuff isn't your bag.

Ico had some interesting ideas that are worth a look.

Rogue Galaxy was pretty okay.

Psychonauts deserves a mention, though it's kind of obvious.

Red Faction had cool political leanings for a game of that era.

You've got heaps of great suggestions already so I'll leave it at that.

Crash Nitro Kart and some of the PS2 Crash platformers are pretty good. The Kart one has a boosting system that I like better than mario kart (and better items and vibes tbh)

Definitely play the AC games. Which series am I talking about? Both.

Armored Core and Ace Combat are really good games. For the PS2 continuity of Armored Core it is arguable that the PSP versions are better due to the extra content, but otherwise it's not too big of a deal.

Primal.

Ps2 Mortal Kombat games were the best, they have beem sliding ever since.

Deadly Alliance is a slog for story/Konquest but a fun game.

Deception is so so good, one of my favourite games ever. Doesnt really hold up as well as Id like but it has brilliant features that I miss in every modern iteration. Namely the best konquest implementation. Chess Kombat.

Shaolin Monks, another one on par with Deception. I have to replay this next.

Armageddon is carnage, just fun fun all the way through. Every character from the entire run of the franchise is playable, it is everythimg.

Future Tactics: The Uprising

its a turn based shooter.

Dark Cloud 2 (aka Dark Chronicle). That and MGS3 are basically the only PS2 games I actively think about still.

Dark Cloud 1 is also worth playing, but I'd play 2 first. It smoothes out a lot of issues with the first and adds so much more to it.

summoner

Colosseum: Road to Freedom

I'm not particularly into sports games but the NBA Street and NFL Street games were SUPER fun. Everything else I loved has already been mentioned in this thread though.