User blog comment:Edwin Shade/A Question Concerning Ordinals (In Particular, Epsilon-Nought)/@comment-5529393-20171219065522

Let's count the total number. The number of times we reduce a successor ordinal is 4^4^4 = 4^256. Next, let's count the number of times we reduce an ordinal of the form a+w^b where b is a successor ordinal. We reduce a+w^1 4^256/4^1 = 4^255 times, a+w^2 is reduced 4^256/4^2 = 4^254 times, and so on up to a+w^{w^3 3 + w^2 3 + w3 + 4}, which is reduced 1 time. Next let's count ordinals of the form a + w^{b + w^c} where c is a successor ordinal. If c = 1 we get 4^256/4^4 = 2^252; if c=2 we get 4^256/4^4^2 = 2^240; if c=3 we get 4^256/4^4^3 = 4^192; if c=4 we get 4^256/4^256 = 1. Finally, the ordinal ordinal left is w^w^w, which is reduced once. So the total number of reductions is:

4^256 + 4^255 + ... + 4 + 1 + 4^252 + 4^240 + 4^192 + 1 + 1

= (4^257-1)/3 + 4^252 + 4^240 + 4^192 + 2.