Every actor to have played the Doctor, with only a couple of exceptions, blend a handful of strikingly handsome or even beautiful physical traits with a slightly larger amount of funny-looking ones. Add in the requisite Adorkable mannerisms, charisma and loveable character acting and you've got the recipe for over 50 years of slightly embarrassing Celeb Crushes. Steven Moffat said every actor to play the Doctor should be "arresting" and "attractive in a very odd way".
The Marshchild from Full Circle, a creature that's essentially a baby swamp monster. In one of the first scenes in which it appears, it's shown peering at the Doctor from behind a fallen tree, its obvious timidity combined with its endearingly ugly face giving it an almost irresistible cuteness. That cuteness is only enhanced by a subsequent scene in which it's shown materializing in a spaceship corridor behind the Doctor, only to go scurrying back out of sight a moment later. It doesn't help that its subsequent ill-treatment by a bunch of (apparent) humans propels it firmly into Woobie territory.
Daleks in general, at least their metal casings. What's not to love about a metal pepper shaker with a plunger and half an eggbeater for arms? Special reference goes to the squeaky-voiced and gurgly humanised Daleks in "The Evil of the Daleks", Dalek Caan and the Ironside in "Victory of the Daleks".
The Ood. Due to their benevolence and the fact they are all Woobies, once you get past the fact they've got mince meat where their mouths should be, they are very sweet.
The tiny old Doctor that the Master kept in a birdcage from the S3 finale. He was so tiny and sad! And kind of looked like Tweety Bird, between the birdcage and the huge head. Also, there's something very endearing about the Doctor's personality in that minuscule body.
The baby Adiposians. Basically a pound of roughly humanoid animated fat. Possibly the cutest thing the show has ever featured. They become less cute when you remember that that pound of humanoid animated fat killed people when they came into existence, although 99% just detach from a human and walk away one by one. The main characters even comment that it's not that bad a weight loss program, all things considered.
The Gangers are incomplete, unstable copies of human beings, all with a smooth, featureless Nightmare Face with visible veins that looks like they are melting. They also managed to look vulnerable and sweet, especially the positively dainty Ganger of Jennifer.
Rex (a Coelurosauravus, a reptile resembling an Indonesian Flying Lizard) and the Diictodon (small mammal-like reptiles best described as reptilian gophers mixed with pigs).
Nimrod, the baby mammal-like-reptile-sea-creature-thing from NBC's short-lived series Surface. Come on, who wouldn't want to love a creature that, when full grown, can swallow a small boat whole, burrow through miles of rock, sunbathe in a magma chamber, secrete a substance that lets it regenerate injuries, and generate bolts of lightning?
The Vork from the Farscape episode "Beware of Dog".
Star Trek: The Next Generation had cute alien critters from time to time. A bug-like species in the episode "The Most Toys" and a thing that Worf's son Alexander becomes fond of in "New Ground".
Alf was most likely designed with these intentions.
It would seem that one of the biggest reasons for the success of The Munsters was their ability to create quaint, relatable people based on classic Universal horror icons. Fred Gwynne as Herman Munster in particular is just plain adorable.
Pigmon, who first appeared in the original Ultraman, a red biped about the size of a child and with a gentle and friendly personality towards humans. In most of its appearances, it helps the human characters in some way, such as warning them team of a monster's arrival, befriending a child, or taking care of people trapped in Kaiju territory. Unfortunately, Pigmon is usually killed by the Monster of the Week while protecting humans.
Eleking from the same series as Miclas is considered cute by Japanese fans, despite being a 55-meter tall mix of an eyeless electric eel, a giant salamander with antennae, and Godzilla. A Chibi version of the monster appears in Ultraman Mebius as the Team Pet.
A lot of the kaiju in Ultraman Cosmos could be considered this, as they're generally Gentle Giants. A good example is Mogrudon, a giant blue mole who often acts like a big puppy dog.