I think that the absolute weirdest was a Nintendo title called City Connection… In it, you drive a brakeless car to leap back-and-forth between platforms to paint the roadway white, while shooting police cars with oil cans and trying to avoid fatal collisions with them or cats. Detailed description & video are here: http://nesguide.com/games/cityconnection/
Oh, and another one is Ladybug, which was on several platforms… It was somewhat like PacMan, except the player could pivot the numerous walls and there was a variety of powerups & enemies (bizarre-looking creepy insects). Condensed ~3 min clip someone posted: http://www.youtube.com/watch?v=F38JJIAr7Eg
Evolution on the Apple ][. You start as a single cell and evolve to a frog, mouse, beaver, and caveman (I think). The final stage you are a super-human fighting robots. The pay off for winning the game is thermonuclear destruction of planet Earth.
The original Leisure Suit Larry on the Atari 800, weird only because I was way, way too young be playing it at the time, and I didn't, um, get it. When you're asking your mom how to solve puzzles like avoiding getting the clap from the ho… that's weird.
Katamari Damarcy (X-box 360)
roll up the world into a giant ball and make planets, stars and other bits of the missing cosmos with it. With added Daddy issues.
Lazy Jones (C64)
wander around a hotel with a bunch of rooms all containing mini games, plus the toilet and the broom closet into which you could walk and take a slash…
Hover Bovver (C64)
Steal -er, I mean borrow- up to 3 of your neighbours' lawn mowers to mow 16 differently patterned lawns. All while being chased by the neighbour in question, and, if you mowed over any of the flowers, the gardener. Plus the dog, who you could sick on the neighbour/gardener to keep them at bay, until your dog tolerance meter went down, in which case said dog would start attacking you. Pretty much any of Jeff Minter's games, actually, but this one sticks out as it's the first I ever played on the C64. Llamasoft inspired me to become a programmer
Sentinel (BBC Micro/C64 and many others)
I…don't think I can describe this one. Best to leave it to wikipedia or some of the retro-review sites, I think.
Parappa the Rapper (PSX)
Living in the PAL territories, unfortunately not many of the stranger Japanese games made it over, but this one did at least.
There was one (well, let's be honest, heaps ) for the SNES too, where you had to draw lines and cut off parts of the screen, with anime style art. Had no idea what the point of it all was but it was oddly addictive. Unfortunately I can't remember its name.
I really liked The Dark Eye for Windows PC. It was a mid 90′s release and a weird trippy game based on the collective works of Edgar Allan Poe. It was 1st person but not 3D, you had to click your way around. It was a beautiful game with gorgeous backgrounds and strange puppet characters stitched in and animated. If you can find it, play it. It was the first game to freak me out. Strange voices are constantly in the background of the surreal world and get in your head late at night in the dark.
I think I remember seeing this advert. As for the weirdest game I've ever seen… it would have to be this PC game from the mid 90s. The thing was that it didn't come with instructions and it was in French, so I had no idea how to play it.
It was a side scroller, and you could change into different things a bit like in Altered Beast, but different. I'm pretty sure they were bio-mechanical.