Another old favourite among cat games is:
Being Good
Doesn't sound much like a game, but the most important rule about Being Good is that the cat should be good in such a way as to cause maximum trouble to its owner who can't however give it a thump because it is manifestly Being Good. We had a cat who would, very occasionally, catch some small, inoffensive and squeaky creature and leave it on the scraper mat outside the door. You know -- those flat scrapers that are rather like a chip slicer, with lots of little blades sticking up? And, of course, first thing in the morning you don't look down as you step out ... This might, of course, be a Real cat's way of food preparation. But we knew, and it knew, that in reality it was Being Good.
And from the bit about cats needing different names on different occasions:
d) it climbed up the loft ladder Because It Was There and then, for some reason, decided to skulk right at the back of all the old boxes, carpets, derelict Barbie houses, etc., and won't be coaxed out, and then when you finally drag it out by the scruff of its neck it scratches your arm in a friendly way and takes a beautiful leap which drops it through the open hatchway and onto the stepladder, which then falls over, leaving you poised above a deep stairwell on a winter's afternoon while the rest of the family are out.*
*All right, not perhaps a name you'd use every day, but best to have one ready, just in case, because when you're leaning against the freezing cold water tank trying to staunch the blood with a priceless antique copy of Bunty you don't want to have to tax the imagination as well.
