My cat also does this, but with just a little bit of claw, so it’s like, “keep petting me, or else“

When I'm gaming my cat will sit on my desk and grab my cheek with his paw to let me know he wants pets. If I don't stop and give him pets or don't give him the required amount of pets it escalates from him gently petting my cheek (hey you idiot, I want you to do THIS to ME), to him petting me with claws out, to him slapping me on the face, to angry headbutts.

If he doesn't want pets but just wants to be with me he'll just curl up in my lap with his head in the crook of my elbow and stare up at my face until he falls asleep. I love him so much, he's the absolute best cat.

My yorkie sits up and wraps both if his paws around my hand to get me to continue lol

Our floofball does that all the time. :)

My 150lb pyr/collie will walk up to people, push his head into their side, and wait for pets.

If you're sitting down, he puts his head in your lap. If you don't begin scratches promptly, he will slap you with a 4in wide paw. It doesn't hurt, but he has NO problem slapping everything in your lap off of it, unless it's food. He doesn't quite BEG, but he will occasionally sideways glance at you and make you feel judged.

My friend’s dog likes to shake my hand off, then nose back under my hand immediately to get me to keep petting. At first I thought maybe I was petting him wrong, but then I realized that he just likes going through the “more” routine.

That's adorable, but it does annoy me a tiny bit when they use their snoot to jack your hand/arm. Some dogs can really pop you like that.