Zoe would nibble gently on your elbow as you sat reading in the chair.
The kitty-cat comes up to the side of the chair, reaches up and very gently taps my elbow. If I’m engrossed in a puzzle even that can be quite a startle.
Time for the laser pointer or jingle balls.
Rocky, my elderly gentleman, will sit in front of me, ears perked, tail going a mile a minute. If I make a motion toward a toy, he will tear out of the room full speed, then come running back, ready to fetch.
Pepsi, my setter-lab cross, will paw at my leg gently and give a quick, sharp, bark. If her foolish hoomin fails to respond, she will jump up and down as though she is on springs, barking non-stop.
Daragan, my big red baby, will lay his head on my leg and give me puppy eyes. However if big sis Pepsi starts jumping and barking, he will happily add to the pandemonium.
My now deceased cat tended toward controlled violence. In the year I lived with someone with another cat the universal signal between then was a good sharp swat to the other’s head.
Sometimes there was bringing something he liked to chase.
Occasionally there would be the getting in front of me and making small runs away as I got closer until I gave in and chased him around the apartment.
Sometimes he would come lie on or right by my foot with a distinct meow. It was his way of calling out the evil socked foot monster for a fight. The meow and distance was distinct from I just want to be nearby and am comfortable enough to get your attention to the fact that I trust you enough to lie belly exposed like my back is broken.
Mostly it was the direct violence targeted at me - pouncing, charging, swatting, or all of the above.
Griffin the conure will do things like grab stuff out of your hand, or throw stuff on the floor in front of you, or bounces up and down like his feet are spring-loaded and makes excited noises. If you ignore him he will escalate, and eventually give you a nip to make sure you’re paying attention.
The cockatiels will crawl up your arm and give you a woeful expression, peck at you a bit, and engage in various poses and body language to get you to pay attention to them.
She changes into thong underwear, brings the handcuffs and blindfolds and asks “Master, is this a good time?”
If either one is still breathing, it’s play time.
Whenever Tibby the Cat sees me walking, he constantly criss-crosses my path just ahead of me. He apparently thinks it’s funny to trip me and spill my coffee. When I’m sitting or reclining, he jumps up before me (usually on my laptop keyboard), stretches out incredibly long and thin, rolls onto his back and demands a deep, long tummy-rub. He’s a petite cat with a shockingly loud, deep meow and he’s quite vocal, carrying on long conversations with me (typically of a mousey nature). Between his purrs, chirps, trills and booming yowls, he gets his points across loud and clear.
When Daisy the Schnoodle wants my attention, she dances on her back legs while plunging into my thigh with her two front paws. She does this while huffing and sneezing loudly and wetly. When my leg is sufficiently scratched and covered in mucous, I give up and submit to her demands. My daughter got worse from Tibby before he was neutered, however. He used to mark her as his territory. Soaking her legs was bad enough, but when she was awoken with a warm spritz to the face I heard, “dad, where’s the scissors!”
When Tibby wants to play with Daisy, he ambushes her with a great leap onto her back from above. When Daisy wants to play with Tibby, she bites his ass.
My dog will jump up or down. Or drop a toy in my lap. Or start running around. My dog is ALWAYS up for play.
Prowler’s favorite game is what I call OMG DON’T SCARE ME!!! REALLY!!! DON’T!!! It’s an easy game for me; I go into the bathroom or kitchen, busy myself and appear unconcerned until she pokes her head around the door and stares at me with her big eyes and her solemn little Keane-child face. I act like I don’t notice her for a minute or two, then whip my head around, fix her with a stare, point my finger at her, and whisper “WHAT are you DOING!?”.
This results in mad skittering and racing back into the bedroom, jumping on the bed, wildly lashing tail.
Then I get to do the slow…menacing…Frankenstein Death Walk into the room muttering “I’M GOING TO GET YOU…” and corner her behind the bed. She explodes in a frenzied arc, flying over the bed into a mad skittering dash, skittering and sliding into the kitchen, wild eyes, tail lashing ensues. Repeat.
I have a cardboard box in my home office that officially belongs to The Cat. Whenever she’s ready, she’ll run in, jump in the box and start scratching. That’s my signal to stop whatever I’m doing, pick up a pen or something and then start scratching it over the surface of the box. When I get near a corner, she will then stick a paw out and bat blindly at it. She’s also worn a hole in one side that she can bat blindly out of.
That seems to be the game: it doesn’t matter whether the pen gets caught, it must be vigorously batted at, but she can’t look at it. When she gets tired, she just goes to sleep in the box. 
Papillon Bunny will throw herself down on her back in front of me and wiggle back & forth, making weird little growly noises. English Setter Sugar will either play bow at me, jump on the bed and spin in circles (I call that helicopter dog) or bring me a toy and lay it on my lap.
They’ll pick up their toys or come to me and whine. If I’m up and walking around, they’ll look at me and bark and try to get me to throw a toy down the hall.
LMFAO!!