Studies have shown that rats "giggle," and great apes laugh in response to being tickled. Most other animals do not laugh the way humans do, so if laughter is the response to tickling, then cats, dogs, and most other animals are not ticklish. If we consider responses besides laughter that are characteristic ways other animals show pleasure, then the answer may be yes — they can be ticklish too.
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Two types of tickling
Scientists have identified two types of tickling. Gargalesis provokes uncontrollable laughter and is the one we most associate with tickling. It involves applying repeated pressure to sensitive spots. If you've ever tried to tickle yourself, you know that gargalesis must be done by another person. It promotes social bonding, and it's possible that humans have this response in order to promote closeness between people.
Knismesis is the lighter touch that produces a shudder or shiver that is often pleasurable. It might cause goosebumps to form or a shiver that goes up your spine. Some describe it as a sort of itchy feeling. This response probably appeared to alert humans to the presence of something crawling on their skin, like an insect that has landed and needs to be shooed away before it bites.
Happy rats "giggle" when tickled
Researchers at Humboldt University in Germany tickled rats in various spots and under differing conditions and measured their responses by the high-frequency tones, described as a sort of giggling, that they emitted when they liked the tickling. They found that rats are ticklish only when they're happy and unstressed. They knew the rats enjoyed the tickling because they kept coming back to the hands that tickled them rather than trying to escape or hide. However, when the researchers placed the rats on a high platform, they were not happy because rats are afraid of heights, and in that fearful state, they were not ticklish.
Mother chimps tickle their babies
The great apes, which include chimpanzees, orangutans, gorillas, and bonobos, all emit humanlike laughter when they are tickled. They are the only species aside from humans that tickle each other.
Chimpanzees have been studied extensively, and researchers noted that mother chimpanzees have been seen tickling their infants, who respond with the same type of involuntary, continuous laughter that humans do when tickled. Chimpanzees' laughter is described as low-frequency, staccato grunts that are similar to the way many humans laugh. Gorillas, bonobos, and chimpanzees laugh a lot during play too, but this laughter is shorter than the involuntary laughter produced by tickling.
Dogs and cats show joy differently
You're not likely to see dogs or cats break into uncontrollable laughter, and if you do, you'd be advised to get your vision and hearing checked. However, if you scratch dogs behind their ears or wherever their favorite spots are, their wagging tails show their happiness. It would ruin a cat's image to show that kind of overt joy, but their prolonged purring gives them away. So, cats and dogs obviously don't experience the gargalesis type of tickling. The same goes for horses, cows, pigs, and pretty much any other animal except for rats, great apes, and humans, of course.
Does that mean these other animals are not ticklish? They're not in the way we initially think of the tickling response because they cannot laugh, but give dogs a good belly scratch, and they will remain on their back for as long as you're willing to continue. For many dogs, one leg involuntarily paws the air wildly while you scratch, as if they're experiencing the shivers and shudders of knismesis. This is called the "scratch reflex."
Cats bump their head against your hands again and again asking for head or under-the-chin scratchings like the study rats coming back to the tickling hands. When you indulge them, cats will move their tail back and forth softly, even nudging you again if you dare to stop scratching. Clearly, cats and dogs do experience the second kind of tickling. Like humans, rats, and probably great apes too, dogs and cats have to be in the mood to be tickled and not when they're annoyed, overly stressed, eating, or chasing prey.