Well, not "bestest bests", but definetely worth to try:
Horror-themed:
1)
Forbidden Siren 2. Seriously, try it! I guess reason it's not mentioned often here is that it was only released in Japan and Europe. Also, it's by authors of Silent Hill 1. I recommend not to bother with Siren 1, because Siren can be too frustrating. But FS2 is one of the most interesting horror games (not only on PS2), imho.
2)
Echo Night Beyond. First person adventure - wandering on the moon station, talking to ghosts. Good when you're tired of action and need some loneliness.
3) You can also try
Rule of Rose for nice plot, setting and music - but it's gameplay is far from satisfying.
Other honorable mentions are Fatal Frame series, Haunting Ground, Kuon.
Action:
1)
God Hand - awesome beatemup with ton of humor.
2)
Killer7 - if you dig surreal movies like David Lynch's Mulholland Dr., Guy Ritchie's Revolver or David Cronenberg's Videodrome - this one if for you. Gameplay-wise it's mix of rail shooter and action-adventure. This game is not for everybody, and don't compare it to later Grasshopper games like Lollipop Chainsaw or Killer is Dead.
3)
Global Defense Force - again, no NTSC release. If you want crazy epic battle with hundreds of aliens - this would be fun.
4)
Shinobi and
Nightshade. If you like pre-DMC slashers (just killing, without crazy combos etc), these two are nice. Pretty challenging and with good music.
5)
Transformers Armada. Came out before Transformers movies. Pretty impressive graphics sometimes.
Now, some honorable mentions: Contra: Shattered Soldier, Blood Will Tell: Dororo, Ghosthunter, Ghost in the Shell: Stand Alone Complex, GunGrave 1,2.
Adventure:
1)
Chulip - game where you must kiss everybody.
2)
Disaster Report and
Raw Danger - surviving during catastrophe. Sometimes as thrilling as horror games.
3)
Shadow of Memories. Heartful adventure with very good plot.
Honorable mentions: Drakan, Glass Rose, Okami, Psychonauts.
First Person Shooters:
Well, emm.. Definetely,
Timesplitters series. Maybe also Darkwatch and Black.
Mecha:
1)
Armored Core, as it was said before. I'd add that some people consider Silent Line being the best. Nexus is also good. DON'T start with Last Raven - it's hard as hell if u don't import saves from older ACs. And _avoid_ Formula Front and Ninebreaker - it was just A.I. testing games that later influenced AC5 etc.
2)
Robot Alchemic Drive - epic Godzilla-like fight between distant controlled huge robots.
RPG and not so RPG:
1)
Shadow Tower: Abyss. What an unique game! Kind of father of Demon's Souls. First-person dungeon crawler with interesting things like guns (!) and cutting limbs of enemies. Released only in Japan, there is translation patch somewhere. Also, very atmospheric.
2)
Rogue Galaxy - one of the most beautiful ps2 titles, imho.
Honorable mentions: Eternal Posion, Odin Sphere, Stella Deus.
Megaten titles:
Well, Persona 3 FES and 4,
SMT III Nocturne, Digital Devil Saga 1,3 and DS: Raidou Kuzunoha 1,2. Quite a choice
Open-wolrd-alike games:
Besides Bully and GTA there are
Yakuza 1,2 and
Steambot Chronicles.
Racing:
There is Downhill Domination,
Outrun 2006: Coast 2 Coast, R: Racing Evolution for arcade style
(+ Tokyo Xtreme Racer: Drift ?). And Enthusia for "realistic".
Everything else:
Iron Triangle and Ring of Red are for strategy lovers. Katamary Damacy and Under the Skin are for fun.