Is there a way to cure excess hair growth on the back that doesn't involve repeated shaving or epilation?

Cure?: Permanent hair removal with laser is the closest thing to permanent removal.

Laser: Laser hair removal can be very helpful at reducing the rate of hair growth. However it takes 3-5 monthly treatments and then maintenance treatments (1-3/year). Know that it is not a cure and hair will continue to grow but at lesser rates the more treatments you get. It is also not covered by insurance and can get expensive. If you do get it done, make sure they are a board certified physician.