forget yes I've been waiting for a thread like this for anlong time.
For flight games I'd say Ace Combat games are great especially shattered skies and one of my favorite flight games is an underrated gem is a game called Sky Odyssey it's like an adventure flight game dodging obstacles, exploring, completing objectives, and scoring massive points to get a rank through some tough as stuff environments.
Racing? Gran Turismo, Burnout, NFSU, HP 2, U2, and MW, and Midnight Club to get the popularmones out of the way. Road Trip Adventure is an unique open world racing game with quests and customization. Downhill Domination is a bike racing game where you down hill scoring tricks and battling other opponents by fighting them along the way. Crash Nitro Kart isn't as good as CTR, but it's still a fun kart racer and you should give it a try.
Platformers are obvious, Ape Escape, Ratchet and Clank, Sly Cooper, Jak and Daxter, Spongebob BFBB and The Movie Game, TY, Pac-Man World 2, do I even need to go on?
Open World games, GTA III, VC, and SA, Simpsons Hit & Run, Spiderman 2, Mercenaries, and True Crime.
Party Games, while they aren't amazing on the PS2 are still pretty fun, Spongebob Lights, Camera, Action, Pants and Ape Escape Pumped and Primed to name some of my favorites.
Classic game compilations? Metal Slug Anthology, Sonic Mega Collection Plus, Megaman Aniversary collection and the X Collection.
Of course if I did every genre then this post would probably hit the limit. Shadow of the collesus, Metal Gear Solid, Kingdom Hearts, Legend of Legaia, Phantasy Star Universe, Drakenguard SSX, Tony Hawk Pro Skater, Twisted Metal, Red Faction I & II, TimeSpliters, Quake Revolution, and I know it's not on the PS2, but it can be played on it so Hot Wheels Turbo Racing beats every other HW game it has fast arcade styled action with cool track designs and sick music and is hella fun in multiplayer.