By all means, offer explanations for those really weird ones (ones you know others will probably ask you about if you don’t offer one…:p…or heck, even with a “ordinary” biting if there’s a story behind it).
My list:
Human
Dog
Cat
Spider
Ant
Bee
Misquito
Rabbit
Turtle - feeding it for a friend. It hurt. A lot.
My list, I’m sure, is pretty sheltered compared to others.
Go on and prove me right. 
Dog - I’ve been bitten twice in particular, both times by German Shepards. In the ass. 
Cats
Rats
Mice
Hamsters
Gerbils
Chipmunk
Squirrel - mine
Skunk - also mine
Horse
Birds - several, but the worst was a Sulpher Crested Cockatoo
Chinchillas
Rabbits
Guinea Pigs
Snakes - biggest was a six foot boa, worst was a Mexican Black Kingsnake - little bugger wouldn’t let go!
Chameleon
Argentine Horned Frog - they have teeth!
Tokay Gecko
Aligator Lizard
Wasp - yes, bitten not stung. I had jam on my finger. The wasp decided to take a bite with it’s mandibles. Quite a pinch!
Hedgehog
Degu
Dwarf Hamster
Working at a zoo does have it’s interesting moments!
Dog - Still have the scar on my calf. I deserved it.
Cat - Playing too rough.
Horse - Was just minding my own business and got chomped on the shoulder.
Chinchila - Tried to pick it up. It did not want to be picked up.
Gerbil - Ditto.
Human - Was fighting the kids older brother. Younger brother jumped on my back and bit me.
Bees sting more than bite, don’t they? Ants SFAIK, can sting and bite, depending.
Under insects I could add flies, fleas, gnats, ticks… heck, who knows how many bugs have bitten me over the years?
I could also add under birds: at least one or two pigeons, parakeets, ducks & geese. Parrots tend to have pretty strong beaks, so I keep my bitable appendages out of range.
I used to catch lizards and they routinely bite. At least one or two snakes, the non-venomous grass-type.
Guinea pigs, hamsters, gerbils, mice, - they’ve all probably gotten a chunk of me at some point.
Do fish count? They sorta suck on you more than bite, but I’ve had plenty of fish nibble at me.
Hermit crabs. I used to go down to the beach and let the hermit crabs nibble at my toes. I guess they find athlete’s foot tasty or something, but they really go to town on the more or less dead, rough, calloused skin.
dog
cat
snake
hamster
bee
rabbit
None of them were major.
Oops, I forgot two:
goat
parrot fish
I think that’s it.
It’s astounding how often these bizarre threads are repeated.
Human
Dog
Cat
Bird
Squirrel
Horse
Cow
Goat
Rabbit
Lizard
Snake
Caiman (young)
Hamster
Gerbil
Fish (various)
You said “animal” so this may not count, but numerous insects (ant, wasp, flea, tick, etc.)
Oh, yeah…and one zombie just a little while ago, but I seem fine for nokjlkjafkladjfklaj…BRAAAIIIINNNNNNNNNNNNS
Cat
Dog
Hamster
Gerbil
Parrot
Budgie
Ferret
Horse
Goose
Peacock
Plus numerous insects.
Since you’ve linked to this, I won’t repeat my whole list of the usual animal bites–I’ll just mention the swan again. I don’t think anyone else mentioned being bitten by a swan.
Story? It happened at a lovely lake in Lucerne, Switzerland, about 20 years ago at Christmas-time when I was a student on my holiday travels. There were a number of swans on the lake and I was feeding them; they were mostly quite tame and you could feed them by hand. As I was holding out a piece of bread to one swan, another, larger swan decided that he had more right to that bread than the smaller swan, and shoved it out of the way. I was not going to reward this bad behavior, and offered the bread again to the smaller swan–and that’s when the big guy grabbed two fingers of my hand with his bill.
I was wearing gloves, so I was only pinched hard, no broken skin.
The usual suspects, plus a gooney bird (Layson Albatross) on Midway. In Nam there was a “Blister Bug”, or beetle, that left a secretion on the skin causing blistering. Had one crawl across my upper lip while I was sleeping. Painful, but it went away in a day or two.
Snake (Checkered Keelback - nasty little fuckers, non-venemous, but the bite hurt like a sonofabitch. Also had a VERY close shave with a baby cobra, which, ounce-for-ounce, is more venomous than an adult)
Dog (my friend’s doberman, we were play rough-housing, and suddenly he went postal. The only dog thats ever done this)
Cat (my kid sister’s persian cat. This one I deserved.)
Mouse
Macaw (“Who’s a pretty poll…OUCH!”)
Monkey (Bad one, this. Happened when I was 4 years old, more than 22 years ago, and I still have the scars. My right forearm got the worst of it, since I was using the arm to fend off the nasty flea-ridden bastard. Gashes from the monkey’s teeth were all the way down to the bone. Totally unprovoked, too. Nuke all the unevolving fuckers, I say).
A radioactive spider (I’m no longer a girl-shy nerd, my webcrawling is the stuff of legends, and once I get a hold of Sam Raimi for getting that wimp Toby Macguire to play me on the big screen, I’m going to shoot this spiderweb up his ass!)
Human (In her defense, she got a little carried away with by my amorous ministrations. Woohoo - hicky city.)
Dog
Cat
Horse
Various inscects
mouse
Guinea pig
Hamster
Goat
Human
Birds (cockatiel , Nanday conure, parakeet, ducks, chicken,etc.)
Tiger
I think that’s it…
The usual suspects, plus:
Praying mantis (it stung just a little, but mostly it startled the hell out of me).
Hognose snake (painless)
Chameleon lizard (one of the small ones that hang around on porches and walkways in the South. Didn’t hurt at all, but he wouldn’t let go, and since it was my chin he bit, I called him my “lizard goatee”.)
Rabbit
Horse (I was being nice to the horse and brushing her, and suddenly she swung around and chomped down on my upper arm. Stupid horse.)
Garter snake
Poodles. Two different poodles, both belonging to my grandmother at different times. All other dogs I’ve ever met have loved me, but Grandma seemed to have a talent for adopting evil poodles.
Ferret. That was the day I discovered ferrets do not like being trapped under the clothes basket and taunted with Chicken McNuggets.
Chickens (some of the hens used to have definite views about us taking their eggs.)
Humans (both my kids seemed to love to gnaw on my chin when they were teething babes, which was fine until that first tooth broke through.)
Let’s see:
Human
Two small snakes (they were just frightened, poor things)
Mosquitoes (Bloody bastards)
Ferret (did my foot look tasty?)
Cat
Several types of birds, nearly every bird I owned, although most were just pinches.
Hamster (fucker drew blood!)
Turtle
Dog (more of a nibble really)
My little brother
little yip yip dogs
mosquitos
fleas
horsefly
girls
velociraptor
(ok, just kidding about that last one)
Aside from some of the ones you’ve already listed: my ex-girlfriend… does that count?
Human
Dog
Cat
Hamster
Guinea Pig
Budgie
Goose
Horse
Fire Ant
Chigger
Mosquito
No-see-um
The fire ants hurt the worst, of course - I had scars for years, I tell you!
The horse was kind of painful, too - sumbitch bit my shoulder when I had my back to him; he may have been trying to eat my hair.
Two dogs
Cat
Gray squirrel
Yellow-naped parrot(really painful)
Several turtles
Dozens of lizards
Hundreds of non-poisonous snakes
Caiman
Alligator
Salt-water crocodile
Hellbender
Amphiuma