For example, 5 is a balanced prime in order 1 because it is the average of the prime before it (3) and the prime after it (7).
There is a highest order a prime could be balanced in: the number of primes that are less than it. The question of whether 5, the 3rd prime, is balanced in order 3 is undefined. So the kth prime could be said to be maximally balanced if it's balanced in order k-1; i.e. if you take all the primes less than a prime p, and that many primes greater than it, and average them, you get p again.
The first 500 primes are each less than the average of the primes surrounding them, and the difference tends to increase, but it doesn't strictly increase. So my guess would be that no maximally balanced prime exists, but I doubt there's a quick proof. Am I wrong?