There are quite a few games we both seem to enjoy. The safe easy one are playing with a ball (either a football or a big gym ball) and playing with large dog sqeaky toy (he is getting near to being able to do fetch with this - if I throw it he will go and get it, it's the handing back to me that we're working on
). He likes the squeaky toy because he can terrorise the other horses with it.
Then we play follow-my-leader. This is simple and just means he follows me over things, under things and through narrow gaps (there are handy trees in the field that grow very close together, as well as logs to hop over and a stream to jump over).
Then we play tag (had a great game of tag this morning
). This is a lot faster than follow-my-leader (which is at a walk) - tag is trotting and cantering - he particularly likes following me when I do sudden changes of direction. He gets very excited and can leap in the air. This was scary the first time it happened, but I think I understand the rules better now - tag is definitely a non-contact game and he keeps a safe distance. ETA I never chase - he has to chase me. It's a fine line between being a playmate and a predator for human/horse games...
Like Pengapenga says, it is quite a privilege that he will choose to leave the herd to play with me (we often go out of sight of them), so it's both fun and a great way of increasing our bond
It is very good exercise for me as I have to run like crazy up and down big hills in the field - it is also useful, as training canter is difficult without a school, so I'm working at adding a voice cue every time he breaks into a canter.