I can only really speak for ps2/psx games, as I didn't pick up xbox till 360 came around buuut there's some good ones I haven't seen mentioned.
RPGs-
Anything Final Fantasy, VII-XII as well as anthology(PSX)(V and VI I believe), collection(PSX)(FFI and chrono trigger?) and of course the great FF Tactics(psx). Highlights would be VII, X and XII, and tactics.
Breath of Fire 3 - excellent and fairly unheralded for some reason
Legend of Dragoon
Shooters-
Red Faction - one of the best console shooters period, 2 isn't as good, but still solid.
I keep hearing Timesplitters 2 as being good, but I could never get into it myself, it's been gathering dust for years.
Gradius V - It's Gradius. A great shmup that will break you.
Ace Combat - Any out of the series, 4, 5 and zero being ps2, the original(air combat), 2 and 3 being psx. Excellent games if you're even vaguely into jet fighter games.
Katamari Damacy and Everyone <3's Katamari - I cannot reccomend Katamari highly enough, it absolutely rules.
Ratchet & Clank - Pick one from the series, they're all excellent.
Armored Core - Be the pilot of a giant robot charging around blowing up other giant robots... oh and you're a merc! Ridiculous levels of customization as well. There's a few psx and a few ps2, just pick one, you won't go wrong.
Max Payne - a fun, dark, and at times hokey festival of blood and explosions.
Devil may cry - I can't speak on the sequels, but the original is definitely fun, even if it's a bit of an emo kid.
Prince of Persia: Sands of Time - rediculously fun and just plain ridiculous acrobatic moves and rewinding time. You can't go wrong. I can't speak on the sequels, as I haven't played them, don't know why I haven't though.
Resident Evil 4 - Pick this up on ps2, or gamecube if you don't have a wii. If you have a wii, get it on there.
X-Com: UFO Defense - If you can find it, one of the best strategy games of all time. It's psx, but psx games run on ps2. It's on PC as well, but you need to find a user made patch to get it to work on a computer built in the last 8 years.
That's all I got.
[edit] - Grand Theft Auto III - probably the best out of the series.