I thought my late cat and I invented the game known as paw paw, but it turns out not. Our variant was that I would try and tap her front paw while she would try to claw me with the other front paw.
I also did something called Cat Identification, which involved tapping the kitty’s left ear, nose, and right ear while singing the NBC Chimes to the letters C…A…T. The fun we had. Sigh.
My cats know how to spell. When it’s time for me to feed them, I yell “E.A.T.” and they run into the kitchen from wherever they are in the house. If they’re close, I don’t even have to yell it.
And one of them plays dead (or dying). He plops down on his back, and rolls back and forth in a sort of twitching motion . . . all the while watching me, to make sure I’m paying attention. I’m sure that his former owner taught him to play dead, but I have no idea what the command was. So it’s now called “happy dance.”
“Let’s put your hat on” - stop and let me put your collar on before we go outside.
“Let mommy fix it” - stop walking so I can get your leash pulling from the correct side or so I can untangle it from your underside.
I have a bunch of them. I tell them (three dogs) No Maypole to keep them from wrapping me up with their leashes. There’s also Stay in the Channel, meaning stay on the damn sidewalk and stop trying run into the street.
Oh, and No Containership, when one of the boys makes a 300 yard wide turn after peeing. Dogs, in my opinion, should have a smaller turning radius than a fully loaded container vessel.
I had to stop using the word ‘walk’, and eventually had to stop spelling it. So I replaced it with a nonsense word which they quickly learned also. But “Doodle” is not a word I say that often other than when I suggest one to the dogs, so we don’t get so many false alarms.
I recently discovered that my rescue dog understands some German. Du bist ein guter hund!
Our current dogs understand “pee” which is useful if we’re going somewhere and we want them to hurry their outside business. The also know that “go up” means to go up the front stairs from the driveway to the front door for preparation for going into the house.
“Just stay” they recognize as a statement that we are leaving and they will be home alone. Unlike “stay” they don’t have to remain where they are, but they don’t follow us downstairs to go out. We didn’t really teach them not to follow us downstairs, they just learned they weren’t going to get out so don’t bother.
Back when YoungGuy wsa little, Burger King was his favorite restaurant. Our previous dog learned what Burger King meant and wold head for the car. She often went in the van with us and got three hamburgers plain (she was a big dog). We had to start referring to it as B K , but she caught onto that as well.
I do “Down in front!” when the cat occults the laptop screen. It never works.
I speak English to Blackjack. As far as I can tell he understands about half of it.
The geese understand ‘Busted’ to mean to get back down into the field. Jezabel also understands 'Busted" to mean to stop whatever she is doing. We will teach the new cat Five ‘Busted’ eventually.
Still trying to lure him out long enough to get a few pictures of him.
My Border Collie (Joe) inherited my kid’s small chair. It had pictures of Winnie the Pooh on it. When he was in the way, we taught him to get on the “Pooh Couch”. This eventually was shortened to the command “Pooh” while pointing at any chair, dog-cushion, car seat etc.
Newcomers thought it was weird when we would point to a chair and say “Joe! Pooh!”. But he understood it meant to go there and lie down.
Some folk have mentioned hand signals and nonverbal cues.
About 3 dogs back I had a big golden mix I had gotten in college.
At around 10 yrs old I realized he had gone stone deaf.
I first noticed this because he would no longer come running like a bat outta hell when he heard his food hit the bowl.
Sure enough, you culd get right behind him and say “Wanna biscuit?” - no response.
But what I found curious was that he continued to “listen” so well - and I ALWAYS walked him off leash.
When I started to pay attention, I was impressed at the amount of attention he paid to me.
Just constantly sending glances my way.
I also realized how much body language - hand gestures, head movement, etc - accompanied my verbal commands.
I also think that by that time he had a pretty good idea of how I wanted him to behave in just about any situation, so he knew what he could and couldn’t get away with.
Really brought home to me how carefully these beasts watch us, and take their cues from us.
“Two minutes”. I say that for any length of time I expect them to tolerate. So, if I’m getting their food ready, I say it to signal hat they will be fed shortly. If I drive to the store it means I’ll be home shortly.
“HORSES”. Means danger. When I’m in the barn and the horses are coming in it is the dog’s warning to be alert. Kinda like yelling, “heads up”.
ETA: “up-up” means to jump into my vehicle. I can have the rear gate open and they do not hop in, until verbally directed.
Our two rabbits understand “Oh boy, oh boy!” to mean that we have something good for them to eat, and they’ll come running to check it out. The crinkle of a plastic bag will have the same effect, as they think it might be a produce bag.
(I also trained them to come when called ("(name), come" accompanied with a beckoning hand gesture) and sit up on their haunches ("(name), up" with a pinching gesture above the head) but those are more typical things you’d train.)
These are sort of standard commands, but our dogs are very good at the RELEASE command (let go of whatever’s in your mouth) and LEAVE IT! (get away from that disgusting/dangerous thing you’re sniffing). I’ve also got Simone somewhat accustomed to GO POTTY – not so much a command as an association, like hearing running water makes one want to go. They also know COME ACROSS, which means to hurry across the street and not linger to stare at cars.
Perhaps the most unusual command we use is PACK! Spoken sharply, it means “this is serious pack business, you need to stop dawdling and obey” and is used to reinforce commands when appropriate.
A highly useful command we have used with our dogs, when they are circling us as we are eating or just being a nonspecific pest, is “GO AWAY”. Combined with a stern tone, this generally results in the beast slinking away with a guilty expression.
I speak some Spanish to our mutt, mostly “Sientate”.
He also knows “get out of the kitchen” (self explanatory) and “towel”, which means to wait until we’ve toweled him off before running into the house.
Just thought of another one - “Hey!”
Especially useful if he looks like he is about to get into something.
Just reminds him that I’m around and paying attention, and keeps him behaving the way expect him to.
They really do study us. Our recent rescue dog noticed the unconscious hand gesture I would make ( I myself was oblivious at first ) when I gave him the “widdle” (pee) command. Now I just have to make that weird hand movement and he knows it’s time to kill some grass.
All three dogs are quite familiar with swear words too. It doesn’t matter how sweetly and positively I say the word “fuck” - they all give me the hairy eyeball and look a little nervous.
Heh. My oldest dog is named Pepita. She responds only to my commands. When someone else tells her to sit, I explain that she is Hispanic and only responds to Spanish. Whatever I say, she does, (via hand signs). 
None of my own, but a friend of mine had a dog once upon time who had learned “Did you flush?” as a command, and would indeed go flush the toilet in response.
One of our cats is deaf, and so everything we say to him is non-standard. Before my now-wife brought Cuthbert into my life, she used to laugh about the fact that she spent more time talking to him than any of the other, hearing-enabled cats. Now, I find that I do the exact same thing. 
On the one hand, it sounds crazy. On the other hand, Cuthbert is precisely as responsive to orders to stop yowling or to get off the counter as any other cat I’ve ever run across.