Most of the one's I've played have been mentioned, but I enjoyed them.
I'd go as far as to say
Amnesia was an 'artsy' game. Though not the most beautiful game to look at, it created an atmosphere that pulls you in and doesn't let go.
Dear Ester was decent, but not really a game. More like an interactive narrative, which made it somewhat unique in it's own light. Enjoyable, but not for everyone.
Mainly, I'd see an 'artsy' game as a game that incorporates it's visual and audio style above and beyond other games. Usually they end up being indie/stand alone titles.
Okami is a big one to come to mind. The visuals weren't just used to be pretty, they were actually a part of the game play.
Viewtiful Joe also did this, but in a more limited fashion.
Yeah, this could get twisted to say shit like "OMG BUT BF3 Has DIRT on your eyes when you get blown up", but it's not really a part of the gameplay.
Braid
SoTC
Ico
Limbo
Okami
Rez
~ I honestly don't think this post made all that much sense, but I hope it does, at least, a little.