(Was "A nice sequence" but that title sucked) Perhaps (A^B)^C = (A^C)^B is the only identity that one needs to worry about generically. If so, then the count for an n-high tower is going to obey the recurrence T(n) = sum(a,b,c with a+b+c=n and a,b,c>0)of 11 * T(a) * T(b) * T(c) with 11 not 12. Except actually, this is not quite the right recurrence. But anyway this one would be soluble using generating functions.