What games have you and your cat co-discovered thru randomness and play?

Not a game per se, "Hugs and Kisses." My cat had to go on a special diet for a few months. I put down her morning food, and she balked at the change. I was leaving for work and feeling like I should stick with her for the day. So I grabbed her, gave her a kiss on the head and, in a silly voice, said "Hugs and kisses, HUGS AND KISSES!" I put her down and she ate her food.
From that point forward, she wouldn't eat her food unless I did the hugs and kisses thing. It had to be the full dance, the same voice, or she wouldn't eat.

One of my cats, despite being an extreme clinger, absolutely will not tolerate being picked up under any circumstance.
Lone exception? If there’s a bug she wants but can’t reach, she will meow until I come lift her up to catch it. One hand under her hind legs, one under her stomach - so she has both front paws free to pin the bug with. Fortunately, she’ll let me summon her too, so whenever there’s a moth or something hanging out on the ceiling, I yell “bug” and the cat comes running to catch it for me.
Right ol’ on-demand vacuum cleaner, that one.

My cat loves to play red light green light. This is something he started on his own and it’s exactly like how we played as kids.
We could be watching TV or doing anything really, he’ll peek around the doorway waiting until we notice him, sometimes “chirping” a little if we don’t notice, we turn and look away, look back, he’s moved a little closer holding completely still and staring at us, this goes on until he finally reaches us and lets out a playful meow and plops to the ground for belly rubs.
We can also start the game ourselves if we make eye contact with him and start looking away back and forth to get it started.

My cat and I play tag. She'll run up to me and just as she grazes me, turns around and bolts the other way. Then she hides behind a curtain or her tunnel until I chase her down and play with her
- I have one that plays tag too. I have to yell out "where's my Donut?" and she pokes her head out around the corner of the room she's in so I can 'find' her and boop her nose

My cat likes to jump onto my balls while I am fast asleep at 4am and then walk up my body to meow directly in my face before she shakes her head violently and gets little cold droplets of drool all over me.
It's just lovely.
I've started "trapping" her under the blanket and wrapping my body around her. She enjoys it for a few minutes before she's trying to escape at which point she leaves me alone until my alarm goes off.

If my cat is chilling somewhere and I walk by I'll do a fist bump and he will head butt my knuckles.
Open hand pays no attention

I think my cat knows it’s a game when I put my hand under the blanket and swish around on the sheets. He especially likes it when the hand reaches over the side of the bed and he can see it move partially and hears that sound of hand swishes across a sheet.
His eyes widen up and he slouches down in attack mode and waits for the kill. But he’s gentle when he goes in for the kill, only grabbing on and then letting go. I give him some scratches and grab him up to hug him close as I scratch him all over his head as a reward.
- Does it not lead to toes snatching when sleeping?
  
  He’s actually really well behaved lol. Today was the first day I’ve had to scold him in a while because he snuck out of the house when I cracked the door open to leave. But he generally sleeps at the foot of my bed right beside or on top of my feet.
  But for the most part, he’s very obedient and kind to me. I taught him as a young kitten to not bite and to instead “ask” for pets. He was an extremely violent kitten! He would always bite. I started scolding him and he would stop and would instead start to just lean in slowly for a bite because he didn’t want to hear me yell and when he would do that, I would scratch his ears or under his chin since his face was already near my hand, which is his favorite place to get scratches. I love touching his paws and he hatesss that. So that was the perfect teaching opportunity, whenever he would lean in to bite, I immediately scratched his chin and he began associating discomfort with redirecting me instead of resorting to biting me. So when he would get those urges to bite, even the love biting, he now just leans in for a scratch and I give him a good scratch under his chin and he goes wild with purring. I also unintentionally taught him to use his paws to grab my hand to ask for pets too, so he does that instead of biting too and will try to get me to not stop scratching his chin. I think that came from me touching his paws and he associated that with scratches so he does that to “ask” for scratches. He often reaches up with his little paws to demand scratches while I’m working from home even in meetings.
  He bit me on my foot once a few months ago but I could tell it was because I was petting him with my foot and he wasn’t enjoying it and in hindsight, I think I wasn’t paying enough attention to his warnings to ask me to stop. I can’t remember the last time he tried biting me before that.
  He was declawed many years ago by my parents when I lived with them, so scratching hasn’t been an issue. That’s the reason I’m very strict about him being outside because we have owls and coyotes and other cats and he can’t defend himself from them like other cats can. I take him outside supervised for a little while each day which has helped our bond too. I can tell he trusts me a lot and will go in when I tell him to even though he loves being outside. So I think he doesn’t want to bite me because of the bond we have too.
- Same, good use for catlike reflexes

My bathroom was right at the top of a long staircase. When I'd be putting makeup on in the morning, Gilligan would bring me one of the small sponge balls from the living room and drop it just inside the door. I'd flick it down the stairs and he'd leap down 2-3 stairs at a time, find the ball and bring it back for the next round. He'd do it over and over again, every day. Miss that boy. Great post!

Me and my cat like to bring each other the successful kills from our hunts, but unlike my cat I stick to my species....
/s
I think one of the more funny spontaneous games that I have seen someone come up with their cat was they found out their cat loves to bat around wine corks on their wooden floor, so they got a big jar of wine corks and occasionally would throw one onto the floor and the cat would launch off like a rocket, batting at it and bouncing it off the walls.
I have seen that kind of thing before with a cat batting an object and chasing it as it richoted across a room, but the wine corks just worked SO well and this cat really really really loved chasing them.
- I have so many toys for that cat. I wish mine were as... let's say passionate about wine as I am.

My cat will bring a toy down the stairs to us and stare and meow until we play with her. It's absolutely heart melting and impossible to refuse. She also (at 7 lbs) will bring slippers, toques, huge rolls of yarn, and toys up to the bedroom at night. Unfortunately for her we do not play since we are sleeping.

Our cat is basically fearless and gets bored easily. We like closing doors in our house upstairs to keep the temperature of the house more stable. Naturally when she is in one of those rooms we don’t want to lock her in so we try to get her out. She made that into a game of chase combined with hide and seek. Eventually that grew into a game she’d initialize throughout the whole house where she’d randomly fake-lunge at us and then sprint off. Or she’d hide behind a corner to ambush us. Sometimes we’d chase her. At other times we’d run away and hide and she’d come to find us and once she found us she’d run away again.
Problem is we’d be done playing after 5 minutes. She’d want to go on for 30 minutes at least. Anyway, she doesn’t initialize anymore since we got her a kitty friend. Now she plays a similar game with him that will sometimes go on for an hour at a time.

I taught my boi to meow at me when I come around a corner or pop out from behind a door frame. Kinda like our version of peekaboo.

Ohhh, we play the bathrobe rope tie too! He likes to play fetch with crumbled receipts, hide my slippers, and sometimes work on adobe audition although sometimes it gets boring.