I got through phases of games, and right now I'm in the one I like to think of everything-modern-is-making-me-angry-so-I'll-turn-back-to-vintage-games - and that's in the form of PCSX2 on my Steam Deck.
Playing the PS2 generation on the Steam Deck is kinda the 'perfect' generation: I can upscale the resolution without a performance hit, I can apply HD texture packs, Retro Achievements and I can drop the TDP to extend the battery life.
Anyway, I never had a PS2 (it was just a touch before my time), so there's so many games I'm yet to discover there. I'd love to hear any suggestions you might have for what I can play on there:
Just a couple notes:
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No GTA games because they're better with the PC versions. While I've bought the games, the account that Rockstar demands people have to play their games makes me angry on principle (a single-player game forcing an online check each week to be able to play it? No thanks), so I've got cracked versions to play.
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No Final Fantasy because I'm not even sure where to start!
So if anyone was around back in the PS2 era, and has some recommendations for games they've loved, I'm all ears. I've been having such a fun time playing these.
Some big thanks to my friends who made and run RetroDECK (my choice for emulating on the Steam Deck), and to the PCSX2 team (who I'm chatting to right now actually, so keep an eye out and in a week or so I'll share that chat with them all about creating and maintaining the PS2 emulator!)
Also, if you do have a Steam Deck, and want to play with the HD texture packs I do, then it's really simple. Just download the pack, and name then according to their game's I.D. code - SLUS-20743 for example, then place it in the texture_packs folder:
From there, go into PCSX2, then right-click the game, select properties, graphics, texture replacement and tick the two boxes I have:
Then you've got the games looking the best they can be!
Here's a link to a stack of pre-done HD textures for a lot of PS2 games, to help you on your way!

I think that you're going to likely get more-helpful suggestions if you list some games or genres that you like, something beyond "No Final Fantasy" and "No GTA".
This Reddit post has a list of PS2 games that "still hold up", without genre restrictions. There's nothing there that I glance at and say "oh, I loved that and one needs to go back and play it", but it's probably a reasonable starting point. Like, I enjoyed Max Payne (which I recommend playing on the PC rather than console) when it came out, but I don't know if I'd go back and play it as an FPS in 2025.
I can understand that, but having never owned any PS2, and only played some of what it offered by modern remakes...there's really nothing I'm going to ignore. Not sure why I'd cull this down when I don't want any suggestions culled down.
What I'm looking for is what people enjoyed playing themselves. Generally when someone loved a game, they'll recommend it and explain why they did. Thanks anyway!
Well, fair enough.
One point that someone does make in that thread where someone also brings up the "where to start with Final Fantasy" is that it doesn't really matter that much, because the series isn't in one universe
it's a bunch of stand-alone games. It's not quite like you're starting on trying to read, say, Hellboy comics many decades into multiple series or something like that. The games did evolve in the technical sense, but you won't ruin a game by playing others "out of order".
Just thought I'd highlight that, since you said almost the same thing in your post.