User blog comment:AJZajac/String Transforms (Outline - more detailed post to follow...)/@comment-25601061-20171016161117/@comment-31837178-20171019033116

I'm building towards that, but briefly the '000'(n) transforms get the critical part of their growth from the application of the '00'(n) transform p(n) times, where p is a polynomial in n. The growth of the '00'(n) transform done p(n) times on the successor function s(n) is roughly p'( ω ), where the polynomial p' is one degree higher than p.  '000'(n) basically diagonalizes this over n, so the growth rate is more like