User blog comment:Vel!/Omega and power towers/@comment-5529393-20130418175057/@comment-5529393-20130418190856

Yes, but the numbers of distinct values are different for finite number bases. (See here)

The finite number case is certainly interesting, though. Let F_m (n) be the number of different values you can get for a power tower of n m's. Excepting the case m = 2 (which seems to be a special case, since 2^(2^2) = (2^2)^2), it appears that F_m(n) = F_m' (n) for m' > m and n <= m+3, and F_m(n) + 1 = F_m' (n) for m' > m and n = m + 4. So the values of F_m stabilize as m goes to infinity, so we get a limit function F(n) where F_m(n) = F(n) when m >= n-3. It also appears that F(n) <= W(n). Quite interesting!