Here's a few starters, so you get the idea. Please contribute!
Of course, we couldn't have a list like this without including Katamari Damacy. I will never forget the time I first put this game into my Playstation 2 drive and watched the opening credits. It was a wonderful WTF moment that sent me on a mad search to find other equally bizarre games.
Katamari Damacy (PS2)
Here's a few other oddballs that you've just got to see if you've never seen them before:
Muscle March (Wii)
And you thought the Wii was nothing but shovelware games that don't take chances! HA!
Gitaroo Man (PS2)
You're a boy with a talking dog that saves the world by turning into a robot guitar shredder superhero. Isn't that just wonderfully normal.
Space Channel 5 special edition (Dreamcast/PS2)
The first Space Channel 5 was really weird with it's retro futurist musical news reporting. This is the game Austin Powers would have made if he was computer programmer instead of an international man of mystery. They upped the ante by getting MJ in the sequel.
Chibi Robo (Gamecube)
One of the rare games where your job is to run around making people happy instead of killing everyone.
Ripper (PC)
Christopher Walken. Need I say more? And yes, there is plenty of cowbell.
Parodius (NES/SNES)
This was a great little tongue in cheek shooter that gave you lots of random weird stuff to fire at. This boss takes the cake though.
Conker's Bad Fur Day (N64)
There are lots of odd and funny things in this game, but helping this bee polinate a sunflower borders on pornography.
