Weird and awesome:

Earthbound : A satire of RPGs. Set in modern day, you are (at the start) an average kid, but after some strange events around you, you soon begin to realize your psychic powers.

Plus you get to beat up hippies.

Rule of Roses: You play as the female lead remembering her life at the orphanage. The other kids are cruel and mean. PS2 title, rare game, has kid-killing in it.

Fallout 1 + 2: Post-apocalyptic future where you can do anything. No railroading at ALL, which is a problem, actually. You can kill kids, do drugs, have sex, and what have you.

Dungeon Keeper 2: You play as the guy who builds the dungeons for the heroes to go through. One of the few games where you are the bad guy, really.

Any WarioWare game: Really short, but definitely weird and interesting.

Kirby Golf: I forget the actual title, but it was for the SNES.

Portal Prelude: A really hard mod for Portal. Really, really hard. It’s got robotic voice acting and follows the (bumumed) events before Portal.

The Monkey Island Series: Ohhh my goodness. I cannot begin to state the awesome.