Egg & spoon (w/t/c, who keeps the egg the longest)
Water in a glass (w/t/c, who has the most water in the glass at the end)
Walking race (who can get from start line to finish line fastest - any trot of canter is a disqualification).
Ribbon pairs (two riders hold a crepe paper ribbon between them. w/t/c until only one team is still holding an unbroken ribbon.
Rider switch (two riders ride as pairs while music plays. When music stops, both have to dismount, switch horses, and mouth the other horse. Last pair up is out)
Bareback ride-a-buck (put a dollar bill under your thigh, w/t/c until only one person still has the dollar bill)
Horse soccer (just like it sounds - you can look it up on YouTube)
Trail courses are always fun, you can get creative with all kinds of things to go over, through, under, around. Mailboxes to put things in/out of. Tarps to carry or walk on, HUGE hula hoops to walk through, hanging curtain of balloons, buckets to drag on a rope, gates to go through... the options are really only limited by your imagination.