I have a few games I play with my lesson students.
1. Man Hunt (kind of like tag)
You start with one person "IT" they have to try and catch someone. When they have caught someone then that person is "IT" with the origional person so now it's two people trying to catch the others. This continues until there is just one person left. When that last person is caught they become "IT" for the next round.
RULES: You have to tag the person and not the horse. You are not allowed to hit. You must keep to the speed determind for the level of riders (If in an arena no faster then a trot). If you go faster then the set speed you become "IT". If you hit someone you are in a 3min (or whatever is determind) "Time out" in a corner with the supervisor.
2. What time is it mister wolf.
You start with one person as the wolf at one end of the arena facing away from everyone else (with room to have a horse go past them). The rest of the people start at the other end of the arena. The free people ask the wolf what the time is, the wolf gives a number between 1-whatever limit is decided (I used 12). The free people must then move that many steps forward, counting the front feet of the horse and stop. When the wolf decides they want to chase the others he/she says lunch time, everyone then runs back down the arena and must touch either the wall or fence. If the wolf touches someone before they touch the wall/fence then they become the wolf.
RULES: You have to tag the person and not the horse. When running back you must stick to the gait determined before hand (Trot or canter). You are not allowed to hit.
You can also do red light green light, tag, octopus, anything you can do on the ground you can pretty much do on horseback.
Hope that helps