Weirdest games?

Cho Aniki... any of them.

The Darius games get an honorable mention but they aren't so much weird as they are just surreal.

 

I guess I'll list some. Earthworm Jim and EWJ2 are both pretty weird. In EWJ2, ants throw babies at you, you have to avoid falling grandmas, and you have to protect cows from getting abducted by aliens.

Some other ones:
Switch for Mega CD/Sega CD
WarioWare (all of the main ones)
Point Blank/Gun Bullet for PS1 (this game has some pretty weird humor)
Katamari Damashii

Many Sega games have weird stuff, like how you can play as the Hornet in Fighters Megamix, or the weird garbage can aliens in Alien Storm.

 

Street Fighter 2010: The Final Fight was a very strange (and challenging) game. It always bothered me that CAPCOM did their best to make it tie in with the Street Fighter franchise despite not having anything to do with it.

 

Drihoo for the Japanese Xbox... you play as a drunken dwarf mining in a tunnel...

 

-as opposed to Dwarf Fortress where you act as the guiding hand of many drunken Dwarves mining in tunnels.

 

Since you have a Saturn: PuLiRuLa

Nekketsu Oyako as well.

 

Liquid Kids (aka) Mizubaku Adventure, Prikura Daisakusen, Ninpen Manmaru - Sega Saturn

 

for Xbox, try N.U.D.E. (the game, not you or your friend)

actually not a bad game TBH.