What's the weirdest game, or moment in a game, that you can recall? Something that didn't just strike you as good or bad, but went beyond that, into the realm of the overwhelmingly strange, very odd, mind-boggling, or just plain weird, that made you wonder if the game designers were on some kind of drug?
For me, probably the Sega CD game "Panic!"
The strangest game I played, by far.

Metal Gear Solid 2, the ending.

Oh, yes. That one left me feeling distinctly disoriented.

The final boss in Half-Life.
What the hell was that thing anyway?

One modification to Half-life 2, I don't remember it's name, but I know I met that guy from Far Cry there, then saw some awesome combine posters... Yeah, it was really "Wtf?" kind of thing.
Oh, and Mistake of pythagoros - HL2 MOD too :D

The whole premise behind Eternal Sonata is pretty weird.

Vangers - One for the Road.
A thoroughly bizarre otherworldy top-down driving/combat/rpg/adventure game by K-D Lab.

Conkers Bad Fur Day. That whole game was one long WTF?

I'd say Planescape had the largest amount of delightful oddities for me. After all, having a floating skull, flaming man, sentient suit of armor, a lawful good succubus that has sworn off sex in your party and a quest involving incinerating various body parts for experience and spells all count as the most unique moments I've had with the game.

Eternal Darkness
Psychonauts

Call of Duty 4 has a scene where a Nuclear Bomb goes off and your player is left to stumble around in the aftermath, half dead until you finally keels over and dies.
I can't remember another instance where a game kills off your character half way through as part of the story. It was weird and quite amazing. It's hard to explain if you have not played it but it stunned me at the time.

The single most bizarre experience in a game had to be Monkey Island 2.
Big whoop.

One word: "Killer7"
Here's an example:
http://www.youtube.com/watch?v=3tj7aBVC1Vw

Psychonauts would do it.

Aye, killer7. Absolutely mental game.
Legend of the Mystical Ninja.
Seaman. Talking to a fish with a man's face. Would have loved to see that pitch.
Katamari Damacy. What is it with the Japanese?