Well, then if the rider is advanced, the I would definitely suggest flying changes - useful in dressage AND jumping. Also counter canter.
What about shoulder-in, turns on the forehand, half-passes, upwards and downward transitions.
Canter to halt transitions (and vice-versa).
For jumping lessons:
put a trotting pole in the middle of the school. Ask your student to trot over it and stop right after it. Get him/her to do this in canter. (Great for horses who rush their fences!)
Also put a small cross pole up and ask rider to turn left or right after the jump.
Put two trotting poles about 21m apart (6 canter strides) and ask your student to count the canter strides between the poles as they ride over them.
If he/he does six, then challenge him/her to do 5 or 7.
Then you can work on a really collected canter and ask them to do it in say 8, 9 or 10 strides.
All these are useful for dressage and jumping. Hope that helps!!!
Any exercise that helps you feel your balance (or unbalance) while riding helps once in a while. If they already have a good seat and if they are riding a safe horse(on the lunge), then do this: tie the reins, take away the stirrups and get them to close their eyes. Then you can feel the horse.
Ideas for dressage: lateral work, collection, extension, changes in headset, flying changes, stopping(always good to work on that), transitions, controlling which leads they pick up(go down the center line and ask for a certain lead).
For jumping: Grids-e.g. 2 stride, 1 stride, bounce, 1 stride, 2 stride, bounce. Many people only do grids where the distances get smaller but changing it up or doing it opposite is also very beneficial. You could also do: 1 short stride(20 ft), 1 normal stride(24 ft), 1 long stride(26-28ft). That teaches the horse to really push from behind and have to lift up and also really helps the rider ride different differences. It's for an advanced horse and rider though. The distances I put in are the distances between jumps.
Some jumping exercises:
"A horse can bring you down your path, but you can't bring a horse down your path."