Don't know if "homemade" games and share-/freeware should be included, though...
One of the most obscure games I ever played was "Mr. Bones" on Sega Saturn. You played a skeleton in many different mini-games, but that game was just cool as sh*t... playing "senso"-guitar before a bunch of hard-rockin' skeletons is one Heavy-Metal-fantasy come trve...
"Superhero League of Hoboken" is another one (yeah, it's on the site, but I never really hard many people talk about THIS one...
)... and it fits BOTH descriptions of "obscurity"...
Then there's "The Persian Gulf Inferno"... used to play that on a friend's Amiga 500 years ago a lot, but when trying to get it as an emu-rom years later I wasn't able to find it anywhere... The game itself is not really bizarre, just not widely known, as it seems.
I could list hundreds of old C64-games probably NOBODY here would've ever heard of, but they're beyond competition anyway