Nibbler from Futurama has three eyes (one of which is on a stalk), fangs, walks like a monkey, only has one nostril, adorably devours whole creatures several times his size, and poops starship fuel. Leela still goes "Awwwww" when she first meets him. The animators were deliberately working with this trope. It becomes all the funnier when he starts speaking with the voice of Frank Welker (although he had provided the noises Nibbler made up until that point anyway).
Also, Amy's pet buggalo, Betsy. Buggalo are giant beetles with cowskin-colored shells that can be milked like cows.
A more recent example: Mr. Peppy. Anybody who says they didn't want to hug him after watching him hatch is a filthy, filthy liar.
Fry's dog, Seymour◊, from back in 2000 is basically the hobo-dog version of Scruffy ("the Janitor"). Never the less, he has the awkward adorableness of this trope. It helps that the episode that introduces him is heartbreaking.
In the first movie, A Sitch in Time, we see the duo's first mission, right after Ron purchased Rufus from the pet store. As he introduces the little guy to Kim, Rufus is a baby; a tiny, wrinkly, shrivelled little thing that still manages to be utterly adorable.
Don't forget Ron's onetime mutated cockroach friend, Roachie. We've all seen cockroaches, so I don't even need to explain where the ugly comes in. However, Roachie still manages to be cute, like a "hard-shelled puppy".
Mr. Crocker. Despite being abnormally thin (to the point where his ribs are clearly visible), having a hunchback, crooked teeth, being bald (his hair is actually a wig which pops off at times when he spazzes), and having his ears on his neck, Crocker has a surprising number of fangirls who find him cute/attractive.
Humorously lampshaded in one episode where Timmy's Dad thinks that Mr. Crocker is gorgeous...even though Crocker is clearly supposed to be ugly.
The Anti-Fairies....especially Anti-Wanda with all her crooked teeth and Anti-Cosmo and his Cute Little Fangs.
Silkie, Starfire's pet in Teen Titans fame. A burping mutant worm the size of a cat should not be that cute.
And Monster!Starfire from the episode "Transformation".
Gorgonzola from Chowder. And that's just to name one character.
The art style of Ed, Edd 'n' Eddy is crude in a way that results in the entire cast being this, to various extents depending on the character.
TheKankerSisters are the most prominent examples, particularly May (though Marie is more popular with fans and often considered the most attractive of the Kankers Sisters).
Squibby from the series The Future Is Wild may well be the model for unconventional cuteness. "He" is the series' resident Non-Human Sidekick and is essentially a land-dwelling Mimic Octopus (who in turn, is a great model for Real Life unconventional cuteness; there's something adorable about a chunk of coral staring anxiously at you) with the personality of Curious George.
Clayface, from Batman: The Animated Series. Yes, he's a psychotic shapeshifting clay monster with a horrible temper, but admit it, when he starts mourning for his humanity and showing off those huge yellow eyes, he's just adorable.
There's also Bud and Lou, the pet hyenas of the Joker. Something about those beady little blank eyes and gaping fangy smiles make them cute, especially when Harley, who they really like, is around.
Iris, from the Ruby Gloom series, is a cyclops but still darn cute. A good chunk of the other characters apply— the bag-eyed banshee Misery, the Emo Teenskeleton Skull Boy, the two-headed Frankenstein's monster Frank and Len. It's a part of the show's whole Dark Is Not Evil aesthetic.
Monroe from The Life and Times of Juniper Lee, being a pug, naturally. (Unfortunately, he tends to be rather arrogant and bossy; June herself says that the reason most people find him cute is because they can't hear him talk.)
Meelo from The Legend of Korra was initially designed as a conventionally cute toddler, but the team fell in love with Ryu Ki-Hyun's uglier designs. A direct quote from SDCC: "Strangely though, the more ugly he was the cuter he became."
Considering how varied most of the residents of Foster's are, there's plenty of friends that qualify for this. Quite possibly the only true exception is Duchess, who is downright hideous in both appearance◊ and personality.
The Ice King. He's an old, skinny man, has a long nose and a gravelly voice, and a shaggy beard... but many fans think he's adorable and want to hug him. Perhaps it's the little beady eyes, the tiny pointed fingers and teeth, derpy personality, and cuddly beard that make him cute? It doesn't hurt that he has a sad backstory and is a huge woobie whose greatest desires are friends and companionship.
Cinnamon Bun. He's a cinnamon bun. But he has a really gonky-looking face and an adorably affable and dim-witted personality.
In one episode, Homer comments that David Schwimmer is "handsome in an ugly sort of way".
Dr. Doofenshmirtz from Phineas and Ferb. He admits on the show that he's ugly, what with his extremely crooked teeth, hunched back and large nose. That doesn't stop fans from thinking he's adorable, or even hot.
Another one from the same show, the Super Computer from "Ask A Stupid Question". It's huge for one thing (to the point where most screenshots show it looming over the gang), speaks in a Creepy Monotone, and is overall really strange-looking. And yet, Isabella seems to think "He's cute!"
Lessee, a lot of characters from Earthworm Jim qualify. Well, there's Jim, bug-eyed and goofy, Peter Puppy (cute in normal form, ugly in monster), Princess Whats-Her-Name (ugly on her home planet, cute on Earth) Professor Monkey-For-A-Head who has a monkey grafted upside-down to his forehead with whom he constantly argues, Psy-Crow with his lack of pupils and his eye-blindeningly yellow space suit, Bob the Killer Goldfish and Number Four, and can't forget Evil the Cat and Henchrat.
Doug Tennappel must like this trope... a lot.
Runt from Animaniacs is pretty much the canine mascot of this trope.
Eventually she stops hiding her face behind her hair, and it turns out it's a rather pretty one. This doesn't change the fact that she has a pair of extra arms that would be too big for a person five times her size, and the weight of those extra arms have given her a permanently hunched-over stance.
Twipsy. He may look like Picasso's rejected sketch but he's so darn cute.
In an episode of King of the Hill Kahn describes Bill this way when getting used to the idea of him dating his mother, "You know he's not so bad, he's like one of those little dogs that are so ugly they're almost cute".
Fly Baxter in the 1980s Teenage Mutant Ninja Turtles cartoon is surprisingly cute for a mutant manfly. He's pretty short, has big eyes, a pretty normal mouth, and wears a silly looking vest. Stockman's action figure, however gave him greater height, more prominent segmented eyes, mandibles, and a torn labcoat.
It also helps how completely and disturbingly adorable this guy is. He's an AdorkableJerkass Woobie. Strange, considering he is (or was) an insane middle-aged man.
Many characters from Superjail! get this reception, when not perceived as being flat-out Gonk. With the nature of the show and varying depictions of them by the artists, they can be both:
Jared is a short, middle-aged and overweight man cursed with Four-Fingered Hands and a long, wide oversized skull. Even so, it doesn't stop him from having fans that adore his cartoony appearance.
Lord Stingray is never shown without his Expressive Mask, and is often depicted with shark-like fangs and bulging eyes by certain artists. While intended to make him look more unhinged, his fans' reaction is more of this trope.
Ash is depicted as horribly burnt and deformed, lacking ears and a nose. Later appearances put emphasis on his wide eyes, giving him a somewhat cuter look as his excitable Man Child personality was developed a little more.
The Twins range from simply looking kind of strange with the Big Ol' Unibrow, to being much more on the visibly Gonk side or to this, depending on the artist drawing them in a given episode. Much like Ash above, the increase in "ugly cuteness" has to do with the later seasons, which at times depict them as seemingly younger and more vulnerable than their initial appearances.
Discord, a mess of random animal parts with a weathered, misshapen pony face, manages to be this. It's probably the voice. Especially now that he's reformed and doing things like a Happy Dance because his friendship with Twilight Sparkle is legitimate and actually squeeing when he admits he reformed specifically for Fluttershy.
The Smooze is a Blob Monster that consumes jewelry to grow and cover everything in his ooze, but he's friendly otherwise, wears a bowtie, a top hat, and a huge goofy grin on his... front side.
Snips and Snails tend to careen between this and Gonks depending on their actions in the episodes they're in.
Then there's Chrysalis, who's basically a locust/zombie/vampire/pony mashup. And manages to look kind of adorable while still being terrifying. Even her changeling army somehow manage to come off as cute.
Molly Coddle of Bump in the Night is a mismatched Frankenstein's doll, yet utterly adorable.
The titular Invader Zim and pretty much the enite Irken race.
In-Universe, Ickis has a particularly bad case of this, as he basically looks a lot like a rather anthropomorphic red rabbit. He's pretty good at spooking humans from out of sight, but when they see him, a lot of people tend to start cooing over how cute he looks. This has caused more than a few "scares" to fail and he has quite a complex about it. Even The Gromble has had a few laughs at how Ickis's looks undermine his efforts to be scary.
The Hub's version of Pound Puppies has Freddy from "A Nightmare on Pound Street", who is even described as "cute ugly".
Wooldoor Sockbat from Drawn Together. He's also the most cheerful and friendly guy on the show, which helps.
Stuart ("Stella") the scrawny cat with missing fur patches from Recess.
The Tick gives us the superhero Sewer Urchin. He has an oversized lightbulb-shaped head (though that might just be a helmet), is clad in two shades of clashing purple and blue, wears a yellow oxygen tank, sounds like Dustin Hoffman in Rain Man, is covered in spikes (naturally enough), and is explicitly stated to smell awful to all of the surface-dwelling heroes due to his home territory being, well, the sewers. In spite of all these things going against him, he is also a kind of Crazy-PreparedGadgeteer Genius who doesn't get enough credit and is oddly charming, as well as considerably more likeable than the JerkassMiles GloriosusDirty CowardBatmanpastiche Die Fledermaus. Inexplicably, Sewer Urchin and Die Fledermaus are best friends, possibly of the Vitriolic Best Buds variety.
Mr. Mumbles, the female cat from Dan Vs.. It helps that she inverts Cats Are Mean, thus making her the exact opposite of her owner.
Imp from She-Ra: Princess of Power combines the traits of a bat and piglet in one package. Also Mantenna could melt quite a few hearts. For an evil organization, the Horde does seem to like cute things.
Steven Universe gives us the Centipeetle. It's a giant, acid spitting, bug monster with nasty fanged jaws and a gem eye inside its mouth. But once Steven manages to tame it in "Monster Buddies" it basically acts like an affectionate, if a bit wild, pet. Making its Heroic Sacrifice later in the episode even more heartwrenching