I just did a quiz and one of the questions was if I had ever petted a stingray. I did and it made me wonder what other animals I got the chance to pet during my life. Turns out it's quite a list...
African Pygmy Dormouse - soft fur, but very quick. I walked out of the room with one clinging to my back (luckily, he didn't jump off)
African Pygmy Mouse - it's hard to tell that there even is something sitting on your hand.
Alpaka - very, very soft, but they don't really enjoy being touched
Asian Elephant - surprisingly soft and warm
Atlas Moth - at the London Zoo, they are very soft
Bearded Dragon - their are spiny, but the beard feels awesome
Black and White Ruffed Lemur - fluffy and their feet feel like warm leather. The lemur was fascinated by my long hair and kept stroking it.
Brazilian Tapir - their fur is coarse, but they really bliss out when they get their neck scratched
budgerigar - a soft bundle of feathers
bumblebee - furry and cute and they don't mind being gently touched at all
Bushy-Tailed Jirds - they hate being petted because our hands make their fur greasy. Very soft, though. Don't pull their tail or you will end up holding the skin of the tail.
camel - that shaggy fur is really warm
carp - koi and mirror carps. Not as slimy as you'd think, but they much prefer nibbling your fingers.
cat - who hasn't. The softest cat I ever petted was a Norwegian Forest Cat.
Chinchilla - they are fur and not much else
Chiton - a marine mollusc, soft and a bit slimy.
dog - Samoyeds are my favourite petting dog, if you don't mind white, long hairs all over your clothes.
donkeys - Poitou donkeys have great wooly ears.
Egyptian Jerboa - both Greater and Lesser. They like running around much more than being petted and watching them is hilarious.
emu - the feathers are very soft, but they don't like being touched
English Park Cattle - a calf at a zoo, it was very soft and it followed me around
fancy mouse - soft and so tiny
frogs - several species, they are slimy but as a child I loved playing with the frogs in our pond (I very much doubt the frogs enjoyed it, though)
Giant African Snail - also much less slimy than you'd think
Giant Tortoise - they love being petted, try it if you have the chance. the skin on their neck is really soft. Hands down the most enjoyable animal to pet.
goat - at a lot of petting zoos, various breeds. Smelly, but fun.
guinea pig - it just wanted the cucumber I was holding
Himalayan Tahr - there was this one Tahr at the zoo who used to wait at the fence for people to pet him and he would just close his eyes and enjoy. Petting Tahrs makes you smell like Tahr, though.
honey bee - furry
horse - I love the smell of horses
Indian Python - totally soft and very warm
land hermit crab - some of them are really hairy, especially after molting
leech - there was one living in our pond and I used to look for him because I loved seeing him swim. He had a velvety skin
Marabou - careful with that huge beak. The feathers are not really soft, except on the neck and head.
millipede - several species. May make your fingers smell a bit weird due to their defensive liquid
Mongolian Gerbil - the claws scratch
Moon Jellyfish - it's a lot less like touching jell-o while they are alive and swimming in the water.
Multimammate Mouse - also known as Natal Rat. They are very, very soft, but they can be pretty aggressive and I've had my worst rodent bite ever from them.
Persian Jird - like Bushy-Tailed Jirds, they are not that much into getting petted.
rabbit - an angora rabbit is about the softest thing ever
rat - not as soft as other rodents. I love it when they put their tail around my neck.
roach - they don't like being petted, but hissing roaches at least won't run away immediately. Their exoskeleton is hard and their feet are scratchy.
Rothschild Giraffe - you can feed the giraffes at the zoo here and so I got to pet a giraffe. I also got slobbered on by it when it wrapped its very long and very purple tongue around my hand. The fur is short and not really soft, except on the nose, which feels like a horse's nose.
sea cucumber - they're velvety and not at all slimy
sea urchin - they have those little sucker feet that will cling to your hand
sheep - a few breeds, always soft and great if you have dry hands (the lanolin in their fur is like using hand lotion)
Spiny Mouse - several species. They really are spiny.
stingray - coarse skin, the eggs are also cool to touch
Syrian hamster - I like the short-haired breeds much more than the Teddies and hamsters are the best rodents to smell
Winter White dwarf hamster - their fur gets greasy from human hands really fast, but they are so soft. And they have those great hairy feet.
woodlice - they were my first exotic pets and I had one that grew particularly big. Their exoskeleton is very hard and their feet tickle.
I'm not sure this list is complete, but the really weird or unusual animals at least are all there. How many different animals have you petted?