Aw man, this kind of thread is right up my alley. My intention is to be informative, so hopefully this isn't considered "bashing".
Pico by Oohwap/Christopher Moore: It's mostly about three or four still images that have crazy collisions and the attacks can instant K.O. There's different versions of the character, including one with a Hadouken, but sadly, the only one that's still online is the one with an opaque background.
Sonso (as well as its variations: Sonso Big, Sonso Invisible, and Cojudin) by Smoke: This is a slightly well-remembered classic from back in the old days. It's a very bizarre Kung Fu Man edit made with random scribbled drawings and some very low quality voice clips. It also sports an OHKO special attack.
Lol Wut by Cadavelico: It's an old KFM edit with random sounds taken from TN's Rare Akuma soundpack, and is based off of that old meme taken from "The Biting Pear of Salamanca". It frequently bugs out in battle, often clipping through the floor or getting stuck during an attack. It makes for a very interesting novelty character.
WIP Diddy by Ravenboy9 (wish I had the thing): Sadly, it's currently offline still, but it's one of the most bizarre edits of Omega Tiger Woods that I have ever seen (it's also not a work-in-progress and has no relation to Diddy Kong). It's a little MS Paint robot thing with random sounds taken from other characters, and along with some oddly changed attacks, can clone itself which was apparently completely intended despite usually being a sign of flawed programming.
Anything by Actarus: He's mostly known for his lovely Peter and Bender characters which have broken palettes, overpowered stats, etc., but he has a bunch of other magnificent wonders, most without the palette issues, but still with most of the other problems.
Anything by GM/Denmou: This is the author responsible for characters like Eraser, Moon Face, Black Pet, etc. The majority of his characters are crazy Kung Fu Man edits, such as MEKA KFM that runs back and forth at mach speed, and Devil KFM which clones itself. His creations are actually pretty fun to mess around with if you are into those types of characters.
Anything by D10D10D10: His characters are absolutely insane, and to be honest, I personally find them to be pretty amusing. His characters mainly include different forms of Homer Simpson and Peter Griffin (e.g. Walrus Homer, Beaver Homer, Barfing Peter, Diaper Peter, etc.) and they all have incredibly overpowered stats which make them as ridiculous and annoying to fight as possible. One of his characters "Cosmic World" is literally just a still image of space that has a massive number of HP (it actually can attack though, so it isn't a punching bag).
Biscuit by SuperJoshua19: This one's kind of an oddball, and is only good if you like insanely cheap characters (God Adon and Nightmare Broly-level cheap). It's a few crudely cropped images of a drawn dog (not sure if it's supposed to be from something) that can defeat some enemies in an instant hit and can pull off some crazy strong hypers. Don't mess with Biscuit.
Hack Jack by Servious: A lot of people seem to know this character somehow, but I'm personally not too sure of its history. It's a Kung Fu Man edit with an odd-looking drawn.... pigeon head? I'm honestly not sure what it's supposed to be, and it also turns into random drawn phrases, and I think sometimes stick figures in some of its sprites. Very similar to Sonso, but not quite as popular.
Samson by Cloudius: A strange MS Paint animal with a triangle face that's also an edit of KFM. Its attacks consist of biting, farting, or turning into KFM himself due to unchanged sprites. The author isn't really known for creations like this, and it isn't hosted on the site he hosts his characters on, so I'm guessing it's a first creation of sorts. I only found it from an ancient mega-pack from Arcadboy (same with HJ, Sonso, Lol Wut, and plenty of others)
Retarded Luigi and Retarded Wario by idiot: These were the first of their kind, and although most people hate them, I find them to be interesting novelty characters (the first versions, anyway). For those who don't know, they're spriteswaps of different characters that use a broken palette and an abundance of sprites and sounds taken from other characters. The author made a bunch of updates to these as well as a bunch of other "Retarded" characters, but to me, they aren't the same. Other people, such as Pgrs111MAGEN hopped on the train with those sorts of edits and essentially sucked them dry, which is when it started to get old.