Misha bows, kneels, waves, shakes hands, balances on a drum, and a few other little cute tricks.
And yes, when he first learned them, he would "offer" the behaviors (especially bowing) whether asked for them or not, hoping for a treat. I had to be very firm about that, so whenever he did one of his tricks on his own, I'd turn my back or walk away.
18 years later, he still does them all, but still only when asked.
One of our favorite things to do together, trick-wise, was just a kind of "invisible leadrope" routine that I now realize looked very very similar to the dance routines you see with dogs...we just didn't have music! He would stand and walk beside me, and move side to side, execute turns on the haunches and the forehand, back up, etc., just by watching me and maintaining what I think of as a "heel" position, except on my right instead of on my left like a dog would be.
These were just the kinds of things we did when he was very young, since Arabians have that first three years to wait before starting under saddle training. I firmly believe that it made for a stronger bond, and gave me a huge headstart when it came time to ride. Later foals that I didn't spend so much on the ground time with didn't come along nearly as quickly under saddle.