We already know about eating their own poop and dragging their butts on the floor.
My cat insists on sleeping on my bed. I frequently wake up to find us spooning. Her back pressed against my stomach. Sometimes she circles me. If I'm laying down and turn my head to the right, she circle around to that side, so she can lay down by my face. If I then turn my head to the left; Again she will circle around to that side and look into my eyes while she lays down preparing to drift off to sleep. Think she's obsessed ;-). I feel kind of stalked. (lol)
Smoking (lol). I knew a couple that trained their dog to bark at one of their friends every time she smoked. They were trying to give her a hint to quit. Even when the friend went outside on the porch to smoke, the dog would go to the window and continue to tell her off. Very funny.
Awww, she works of the tuna pounds before she even eats it. I had a cat that would come to me and put up his front paws for me to pick him up; just like a child would do. When I did, he would reach up a nibble on my chin. He always wanted to bury his face in my neck/ sometimes would curl around the back of my neck like those mini fur wrap things.
The dogs run in circles around the living room when they get excited. When go home and they see me they freak out and run in circles really fast until they tire out and then they jump on me and lick me to death.
He's wants to play in water. You need a little pool for him. Ever heard about the dog that was best friends with a large pond fish. A coi? Is that how you spell it? Well, anyway, he would go to the edge of the pond and the fish would come to him/kiss his snout. Sooo cute. When the owner moved, she built a small pond at her new home, so fish and dog could continue freindship. Amazing.
I've probably seen stranger, but one strange behavior that immediately comes to mind is that my daughter's (and sort of "my" at this point) cat seems to try to bury her dinner, as if she were bury muck in her litter box. She has a nice, clean, dish; and the food will be brand new; and yet she'll be working away as if she's trying to bury it. Then she'll eat it.