Weird as in animation style, plot or characters.
My two are Bobobo-bo Bo-bobo(anime) and Little Bastards (Spain).
Porco Rosso is hard to beat…
Ralph Bakshi’s treatment of Mighty Mouse (with guest appearances by the Mighty Heroes!) was outré to the ultimate! Lovely stuff, but, by gar, it was weird!
Speaking of Bakshi, “Cool World” had plenty of weirdness to smear around. And, come to think of it, “Wizards.” Guy has the word “weird” branded on his private parts, sweartagod.
I once had the joy of seeing a showing of unfinished animation done by the students at Cal Arts (California Institute for the Arts.) This was a whole bunch of short work by animation students, in various phases of completion. Some was just animated pencils, while others were final products. The weirdness was utterly radioactive.
Fantastic Planet
Ren and Stempy’s Adult Party
Speaking of Bobobo there was an obscure anime called Sexy Commando which is where I think they got the basic idea from.
I always thought Lupo the Butcher was pretty out there. And I’m not sure if this qualifies as a cartoon, but it is seriously weird. It’s animated horse racing on DVD; I think the idea is that you’d put the DVD in the player and it would pick a track at random, and you’d bet on the outcome of that race. That’s not the weird part. One of the horses is on his hind legs and running sideways, and that’s not the weird part, either.
Dudu the Floatie aka Puka Puka Juju (anime).
You’re right - who on EARTH wears purple silks? That’s weird!
I’d throw out Bathtime in Clerkenwell as contender for Pretty Damned Weird. (Earworm warning.)
Most? There are a heckuva a lot of strange, psychedelic animation, much too many to list here.
But most? Hmm… Anything by David Firth is a good contender.
Gandy Goose: - YouTube
Seconding Bakshi’s Mighty Mouse.
Rob Zombie’s The Haunted World of El Superbeasto
It might not be The Weirdest, but it certainly is weird and totally worth seeing.
Walter Lantz’s “Boy Meets Dog” is pretty strange. There’s another short on the same tape I had with literary characters coming to life.
Love that one. Saw it on Showtime Shorts back in the '80s.
The first cartoon that came to my mind is The Brothers Grunt
MTV’s Liquid Television (yea, I’m old) aired The Maxx waaaaay back in the last century. I found it pretty weird. Also, at about the same time, was Spike and Mike’s Sick and Twisted Animation Festival. I went every year it came around. Which made me one of the first to see Bevis and Butthead. They were pretty tame compared to some of the other shorts shown.
Hey look. Sick and Twisted Animation Festival has a wiki page. Not sure why I’m surprised. Everything has a wiki page. Alas, there are no links to the individual shorts. But believe me, look some of them up. Really, really weird. In fact, it was at one of these showing were I first heard of (and saw!) a gummer.
I’d nominate* Worker and Parasite*, but the stuff it was spoofing **(the Czech-produced Tom and Jerry cartoons from 1961-62) was often freakier.
For mainstream fare that has coherent story/art but is nevertheless disturbing psychologically, there’s Warner’s Chow Hound from 1951.
The works of Don Hertzfeldt.
The works of David Firth, most well known for Salad Fingers.
Aeon Flux and The Maxx, shown back in the '90s on MTV’s Liquid Television, might be the weirdest mainstream American cartoons I’ve seen. Aeon Flux was particularly good. I see Biggirl already mentioned The Maxx. Do you remember The Head? About the purple alien who lived in the guy’s giant head? And the two gay government guys who chased them? Or the guy with a lawn mower blade embedded in his skull?
John Kricfalusi is well known for Ren & Stimpy, but produced a lot of other off the wall stuff. I still have yet to see The Ripping Friends, but I hear it’s pretty weird.
I didn’t list any anime because that’s easy mode.
The Dot and the Line: A Romance in Lower Mathematics (1965)
by Chuck Jones.
I was going to mention Aeon Flux but thought everybody at least heard of it, what with Charlize Theon playing her in a movie. But that was also a while ago. I forget how old I am and how short memories are.