A.k.A.
Both, classics and new. I like pretty much everything except first person shooter/war/sport/racing.
Ikaruga is an interesting arcade style shooter for the Gamecube. Like 1948 and those old airplane games.
Ico is one of my favorite puzzle/adventure games for the PS2. It just looks really good, and it's really kind of bare in it's story so I appriciate it. There are only maybe 4 story scenes in the game, but it's really cute.
Mike Tyson's Punch-Out was a good arcade boxer for the NES. I still like to go back and play that from time to time. It's a good timekiller.
The Phantasy Star Collection for the GBA is good. Phantasy Star I, II, and III. But no IV sadly. I is my favorite, but III has a really good "Generation system" where you play the main character's children, and even grandchildren.
Old Megaman games are fun. You can find a collection on just about every system except X-box.
Gitaroo Man is REALLY cool music/rythem game for the PS2 if you can find it.
I've had a lot of fun with Donkey Konga: Jungle Beat for the GC. The bongo controllers are fun to use.
Super Metroid and all of those classics are good bets. I really like how Nintendo is rereleasing a lot of classic games on the GBA. Classic Collections are usually good bets to go with.
And if all else fails, there is always Mame and imports. (Seiken Densetsu 3, Naruto Taisen 3, etc.)