You could do a trotting barrel race (same with pole bending).
My barn did a race where everybody lined up, walked as quickly as they could (without breaking gait) to the other end, trotted back as fast as they could, cantered back down the other way, and then had to get off and lead their horse back as fast as they could on foot. You could leave out the canter part if you want max participation, I suppose.
There’s also that bribe your horse game (I’ve never done it) where you have to get your horse to follow you through a pattern (or obstacles?) without being on a leadrope.
I’ve seen people do things where two riders hold a ribbon of tissue paper between them and have to ride around at different gaits (or through a pattern too? I’m not sure) and the pair that does it the longest without breaking their ribbon is the winner.
Some kind of flag race (again, you could have them do it at a trot or have one for trotting and one for people who want to be able to canter) where they have to pick up a flag (or a stick or whatever) from a barrel on one side of the ring, ride down and around to the other side, deposit it on another barrel and whoever does it the fastest wins.
Generalized obstacle course. Doesn’t have to be fancy. Like walk over a tarp. Navigate through a ‘maze’ of poles on the ground. Have the rider open a gate, ride through, and close it again without getting down (you can do this with a rope looped over a jump standard). Have them pick up a jacket and put the jacket on while the horse stands, then take the jacket off and set it back down again. Stuff like that.
And there’s always costume class. 
EDIT:
And ride a buck where everybody sits on a dollar bill and whoever gets through the class without losing theirs gets to keep the bills.