Hatoful Boyfriend was a trip but it doesnt feel like it's in the spirit of your question. Its indie and was made to be weird.

Takeshis Challenge for the NES got a real official release and was very strange, including an ending that makes fun of you for even sticking around. Ecco the dolphin for the genesis was also pretty weird in terms of plot.

But probably the singularly weirdest game that was widely released was Seaman for the dreamcast. You raised these little tadpoles that grew up into man faced fish, who abused you verbally and were quite spiteful of the player. They also flung their feces out of tubes on their heads.