In no particular order,
1. The Secret Island of Dr. Quandary (DOS, '93)
In this educational game, you're turned into one of 3 hideous dolls, move tires around, and...win tin cups of gobbeldy-goo?
2. Maniac Mansion (NES, '87)
Save your girlfriend from an evil scientist! Oh, and meet up with rock music-loving sentient tentacles and potentially microwave a hamster.
3. Donkey Kong Country 3: Dixie Kong's Double Trouble! (SNES, '96)
The other games were kind of weird, but they had a certain flow to them that made the things within their worlds...uh, normal? The first one had you fighting to get back your hoard of bananas. Simple story. You were in the jungle, and then traveled around the world to get your fucking bananas back, helped by various members of the Kong family. Somehow it made sense. In the 2nd one, you were on a pirate adventure, which made sense because it continued from the first. This game is pretty weird on its own (going into bee hives? Riding roller coasters?), but it wasn't such a departure from the first as to be jarring, and it was an awesome game anyway.
The 3rd, on the other hand, seemed to take all the planned normalcy and throw it out the window. Nope, you're not in the jungle anymore - you're in America! Or perhaps the Swiss Alps. You're not just helped by monkeys anymore - now you've got Bear friends! And one of them has clinical depression! You don't fly around anymore - now you have BOATS! (Oh, and Funky decided that he's a hipster now.) Play as a toddler! Fight an octopus in a waterfall as an elephant! Play Simon with crystals in caves to get...Banana Birds! Watch as Wrinkly Kong plays N64, works out with Richard Simmons, and sleeps in her chair! Fight a giant sentient barrel by making her/it belch itself off a ledge!
What the fuck. O_o
4. Earthworm Jim 2 (SNES, '95)
"Anything But Tangerines." Flying in a giant snot balloon. Jim turning into a blind salamander? Weird.
5. Pikmin (NGC, '01)
You play as a teeny-tiny person who has to use fucking sentient radishes to do your bidding. What the fuck. (The weirdest part is that no one seemed to think this was strange when it was released. Oh, radishes? Sweet! They come in different colors? Awesome! They die in battle for you and having fucking radish souls!?)