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.

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.

Ratchet and Clank 3 is missing.

My entire childhood was Tony Hawk’s Underground 2 on the PS2.

The Star Wars Battlefront games. With how horribly the liver service EA version that came out in 2015 was and how dead it apparently is, the PS2 games are the only ones I’ll ever acknowledge as existing.

Also, one of my personal favorites that I’d consider a must have is Dokopon Kingdom. Yes the remake/remaster exists, but I don’t normally go for remasters in most situations. Gameplay is just roll a spinner, move on board, and adventure around the world while trying to do quests if you are in the story mode.

Both games have a lot of content and replayability in my eyes, which makes a game more of a must have for me.

Edit:

Hot Wheels: Beat That! Looked it up because I couldn’t remember the name, but that’s a racing game with a good amount of replayability and isn’t as short as something like Cartoon Network Racing. Didn’t realize it came out on various other consoles and PC, but can definitely recommend on at least PS2 (how I ended up playing it).

There are two amazing super hero games that I would recommend: Spider-Man 2 and Hulk: Ultimate destruction.

Spider-Man 2 was easily one of my favorite games of all time. This game essentially set the bar for what a Spider-Man game should be.

Hulk: Ultimate Destruction was an open world game where you were the Hulk and did all the Hulk things. I can’t think of a recent game where you play as the green giant.

Def Jam: Fight for NY, dot hack series (originals and G.U.), King’s Field 4, Monster Rancher 4, Midnight Club 3 DUB Edition, and quite a few more. I can’t look at the games I have installed just yet, but I could always recommend more if anyone is interested. :)

Try Dynasty Warriors. It’s a genre I don’t see much anymore, but you take control of a general and blast through tons of minions in a big battlefield, aiming for enemy generals to turn the tide of battle (reflected in overall ‘morale’).

All the games tell the same story of the Romance of the Three Kingdoms. Obviously just play the last one available, because it has all the quality of life improvements, but I’ll only play 3, because it’s the only one that has Gan Ning with his super-pimp sword resting on top of his super-pimped no shirt tats.

Ratchet and Clank games

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

Baldur’s Gate Dark Alliance 1 and 2, Champions of Norrath, Star Wars Battlefront 1 and 2, Burnout 3, Jak and Daxter, Rachet and Clank, SSX3, Twisted Metal Black, Tony Hawk Pro Skater 3. I am sure I forgetting some other amazing games.

Dance Dance Revolution!

GTA San Andreas (also GTA III, VC, VCS, LCS), God of War 1/2, Gran Turismo 4, Katamari Damacy, SSX Tricky, Mortal Kombat: Shaolin Monks, Downhill Domination, Tony Hawks Pro Skater 3, Ace Combat 4/5, Resident Evil 4, Metal Slug Anthology, Metal Gear Solid 2/3

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.

Manhunt 1 & 2, God of War