Rock Lee and Guy-sensei from Naruto (aka Thickbrows and Superbrows).
Kisame's sword Samehada, especially when it's cuddling up to Killerbee.
White Zetsu and his brother Guruguru have this going for them. Despite being villains, both are quite childlike to the point of being Affably Evil. WZ only has half a face, but likes to joke around while Guruguru is a humanoid mass of tendrils whose obsession with poo keeps him from seeming sinister even when he is being sinister.
Some of the Tailed Beasts fit this category during that brief flashback where we see them as babies.
Also Sloth. He's huge. He's lumpy, fanged, and freakishly dangerous. He hosts blank eyes and a vapid expression. But something about his unmotivated, drag-your-feet mannerisms and simple speech ("Mendokusai...") make him look like a perfect candidate for a plushie. His big muscles just make him look huggable. His asymmetrical design brings to mind tears like a cheetah's face. He says, "Buh?" and the readers squeal, "So cuuute!" He gets spartan-kicked off a tall building into the snow after wandering around aimlessly in a military base...the fans let out an, "Awww!"
Quite a few characters from One Piece would qualify, especially Usopp, Brook, and Bearsy.
Oars Jr., despite his meteoric apparition, would qualify just in the same way as King Kong - A giant, green, gargantuan (he is at least three times bigger than your average giant!), demon-looking pirate with a soft heart (A typical run-of-the-mill Gentle Giant). He is shot, hit with a giant shockwave, had his leg severed and his heart was impaled by shadows while trying to save his friend who made him a hat that protected him from the rain.
And Marine officer T-Bone, who looks like a zombie despite being alive and well. But he's so compassionate and earnest in his desire to help people that he has the undying loyalty of his subordinates (even if they're a little creeped out by his appearance) and it's hard not to feel bad when the Straw Hats have to plow through him on their way to Enies Lobby.
For a female example, we have Kokoro's granddaughter Chimney.
Pen-Pen from Neon Genesis Evangelion. Seriously, it's not the prettiest penguin you've ever seen (he's got clawed wings, dear God!) but what counts is how endearing he can be, especially when you know why Misato bothers to keep him despite his flaws (like being a glutton, good-for-nothing bird).
Gabumon is one of the early Ugly Cute Digimon. He's a weird lizard-dog-unicorn thing, but you just want to give the guy a hug.
Guilmon would fit here too. In reality, he would look far more terrifying than cute, but his childlike demeanor, antics, and voice just make him so adorable.
Renamon applies as well. The species is described in canon several times that they are tough and intimidating in appearance (especially their glaring eyes) that other Digimon become nervous or threatened by their presence despite the Renamon not intentionally trying to scare others off. This doesn't stop people, be it in the canon or in the fanbase, to still consider the race (especially female ones like Rika's Renamon) to be beautiful, elegant, cute, or (depending on who you ask) sexy.
Even better, L. The manga's artist, Takeshi Obata, said he was supposed to look ugly. And L has legions of fangirls (many of which compare him to a panda!). Doesn't help that the few times he smiles are genuinely endearing.
Rem is also actually quite lovely in an odd way.
Gelus looks like an abandoned rag doll. Awwww!
What's even more awesome is that in the originl designs, Gelus was supposed to be an extremely beautiful Shinigami. Ohba, however, decided that making him The Grotesque would be better because Gelus is such a pathetic figure and the audience would relate to him better. He was right!
Some fans view Hideki Ide falling under this due to his... different style of eyes.
The snail youkai from Karas that accompanies Nue is a bit ugly, but also utterly adorable.
In-Universe example: Nyanko-sensei in his Maneki Neko form from Natsume's Book of Friends is constantly referred to ugly, "pig cat," "world's ugliest cat," etc. by nearly every character who sees him. Taki, as well as much of the fanbase, finds him absolutely adorable. The former's fangirling of him is seen as very odd by her peers and has become something of a Running Gag.
Beerus, the main antagonist of Dragon Ball Z: Battle of Gods. His playful, cat-like nature is hard not to find endearing, and his appearance is based on a Cornish Rex◊, which would also fall into this category.
Fat Buu, post heel face turn, goes from being a vinyl toddler-minded murderer to a heroic pudgy-faced well meaning manchild. His genuine naivete helps here.
Hell, Kid Buu is kind of cute.
Krillin is short to the point of dwarfism, has no nose or chin and when he does let his hair grow out, it makes him look even goofier. Despite this, he's considered endearing enough in-universe that Android #18 gives him a playful peck on the cheek. And then they get married and have child.
Is Seto Kaiba's favorite card, Blue-Eyes White Dragon a "cute" monster? No, but in the episode of Yu-Gi-Oh!, where Yugi duels Johnny Stepps at Kaibaland, there's a scene (cut from the dub, sadly) with a prize machine full of plushies shaped like Duel Monsters, including an adorable BEWD plushie.
Yugi's Kuriboh plays this Trope straight, as do all of its variations. (Or maybe its simply just cute; fans are split.)
Tomoko from Watamote. Admit it, her eyebags make her look cute.
Aikuro's teacher disguise in Kill la Kill, with oversized sunglasses, bad posture, and scruffy hair/stubble. It's supposed to make him look like a plain middle-aged man, but some fans actually find him unbelievably sexy in it, and prefer it to his normal bishounen looks.
Senketsu manages to be one of the most adorable characters in the series despite being a sapient and rather creepy looking school uniform.
Tsuyu Asui of My Hero Academia is a frog girl with huge wide eyes, a wide mouth that is usually in a wide line or smile, long, semi-slimy looking black hair that sometimes blocks her eyes in a Snape-like fashion and of course, frog-like limbs and a long frog tongue. And yet she's beloved by fans because of how adorable her resultant appearance is, obviously the fact she is one of the nicest characters in the show and her Badass Adorable status helps quite a lot.
Todoroki's scarred half of his face is oftenly portrayed like that, particularly by Midoriya/Todoroki shippers.
Iggy from JoJo's Bizarre Adventure initially appears a regular Boston Terrier with bad habits. After the heroes are able to tame him, however, he's very much a playful, helpful, and loyal dog. Subverted later on as Art Evolution tries to draw him with a human-like face, though his new look still is of a prankster.
Bleach has Dondochakka and Pesche, both of whom are quite Gonk-y in terms of character design, but are nonetheless quite lovable.
Liltotto Lamperd is an inverse example. She's adorable by conventional standards, but has an ability that involves distorting her mouth into an abnormally wide maw.
Attack on Titan: A few Titans manage to fall under this, in particular the now memetic "Kawaii Titan".
While most of the characters from Midori No Makibao is gonk, Makibao himself is cute- if you think of him as a hippo instead of a horse.
Shaman King: Chocolove is infamous for being a case of Values Dissonance, but if you look past that, he's pretty cute. In fact, his constant use of terriblepuns is borderline Adorkable. It kind helps that he looks a bit like the aforementioned Usopp.
Chu-Chu from Revolutionary Girl Utena has often been described as an ugly-cute monkey-mouse. It's a balancing act; the design itself is pretty ugly, but the way he acts can be downright adorable (or irritating; individuals' mileage may vary).