Make sure it has a really good stop on it. Voice commands are nice--especially a "whoa" command. Try different patterns using a variety of circles, straight lines, stops, turns, backing up. Just make them up and use various gates. Can you pick up something from off a barrel, sidepass in the middle of the arena or over logs, walk over/in between a flour lines placed on the ground, mount/dismount from the 'wrong side' of the pony?
That being said, a good set of basics on the horse/pony will let it be able to do just about anything.
As for a game, I remember a few 'odd' timed events we used to have that were different. The key to most of these was having a horse that was calm and that would listen to the rider. Although these listed below are timed events, they show some basics that every horse needs to know and they don't have to be done with speed. Just try walking/trotting through some of them.
1) A keyhole race--a flour keyhole type shape is placed at the middle of the arena. Competitors run to the keyhole, enter the 'lane' of the keyhole, stop and turn a 180 in the circle type part and exit back out the lane and race to the timer line. A 5 second penalty each time the horse stepped out of the flour markings. Some horses would not like the flour on the ground and wouldn't go in, others couldn't get the horse stopped in time and basically ran through the obstacle (which was disqualification). Some shows required you to stay in the circle for 5 seconds(a timer person would tell you when to go.) This is one where the 'slow show horses' actually did better than the gaming horses sometimes.
2) Quadangle race--basically 4 poles or cones set up in a square pattern. You race down the middle, make 4 left or 4 right turns around each corner of the square, and race back through the middle. (Setting this one up larger can help getting the horse to do nice turns/circles for patterns.)
3) Egg and Spoon. Not a race, but more in the rail category. It tests how smooth your horse is and how smooth your seat/hands are. Basically go through the different gaits of the horse while carrying a boiled egg on a spoon--but you can't touch the egg, just hold the end of the spoon. (Slow, smooth show horses usually took this class.) If you can do this at all gaits, try circles/patterns.
Trail obstacles would also be something to try. Logs, mailbox, bridge,gate, etc.
Again, I can't emphasize enough, make sure it knows the basics and knows them well. Especially the stop. (I had a young horse I was working with several years ago, went to a show and it panicked in a class and tried to buck. Everytime it did I said "whoa" and it would stop for a bit. (We just had to keep 'stopping' until the announcer was able to get the people to stop racing and yelling on the right on the outside of the arena during the 2 year old class. After that the horse was fine and we weren't the only pair having trouble--another pony started running during this. Some people...)
