User blog comment:Nedherman1/My Factorial vs HAN!!/@comment-5529393-20180703225826

I'm unclear on what you mean by "When it becomes 1, the "n" of the left is reduced by 1, and the process returns." Do you mean the "n" on the right? But 1^^^...^^^n is always 1, so iterating using this number does nothing. Anyway, I'm just going to assume the process ends at 1^^^...^^^n.

[n;n] grows at roughly f_{w+1}(n^2). The function n^^^...^^^n with n arrows grows at the rate of f_w(n), and you repeat this process mn times, or n^2 times when m=n, so we get a little below f_w(n^2).

Then [n;n]^(n) repeats the above procedure n times, so we get roughly f_{w+2}(n).

This is very far from HAN; even the basic array notation for HAN goes up to phi(w,0), which is much greater than w^w^w^w^..., and hence much greater than w+2.