At my barn, we sometimes play the "Sweetheart" game. It requires two horses and a length of rope. Each rider takes an end and the point is to see how long you can ride without either rider letting go of their end at different gaits. It's got a lot to do with maintaining a certain speed and steering. It's pretty fun!
I do this with my students, but use toilet paper. Makes it extremely challenging!
Also use tp to play the "mummy game" I hold the loose end and give the roll to the rider, the goal is to maintain a perfect circle around me and "mummify" me without breaking the tp. The little ones love it, they like anything that makes me look silly really lol.
Pool noodle jousting - or shoot the rings. I have hula hoops on the sides on jump standards and they try to put their noodle through the hoop. I also play joust with them riding.
Egg & spoon is always fun. It can get pretty tough, spoon in mouth, no stirrups, hand gallop, etc.
Ride a buck. Put a dollar bill under rider's thigh bareback. If they make it through all gaits and whoa, back, etc. I let them keep the dollar.
Suitcase race - get a bunch of halloween costumes or really tacky clothing from a secondhand or vintage shop. Race down on horses, put on new outfit in suitcase & race back.
Double bareback (provided horses are tolerant and riders able) is always fun, especially if you require them to switch positions without getting off the horse.
When it's hot, getting water involved is always fun. 2 barrels, 2 5 gallon buckets. Give a certain amount of time and whoever can transfer the most from their full bucket to the empty via 8 oz. cup wins. Put disposable aluminum pie tins on jump standards or fence posts, squirt gun and play cowboy mounted shooting. With the tins you can hear if it's a hit or not.