User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-31966679-20180623154722/@comment-35470197-20180627015339

Ok. Then the new transformation \(a(n) \mapsto a^{a^x(n)}(n)\) is much bigger than the previous \(a(n) \mapsto a^{x!}(n)\).

On the other hand, when you compute \(a_{b+2}(n)\) in your previous definition, then you will notice \(a_{b+1}(n) < a_b^{a_b^n(n)} < a_{b+2}(n)\).

Therefore your new definition of \(a_b\) is bounded by your previous definition of \(a_{2b+2}\). So it is roughly bounded by \(f_{\omega + 4b + 4}\). In order to go beyond \(f_{\omega b}\), you need additional methods.