User blog comment:Eners49/New Notation Idea? (Attempt 2)/@comment-25601061-20180712033049/@comment-35470197-20180712222827

@Eners49

\begin{eqnarray*} f_2(a) < & \{a\} & < f_3(a) \\ f_3(a) < & \{\cdots\{a\}\cdots\} & < f_4(a) \\ f_3(a) < & \{a,2\}& < f_4^2(a) \\ f_3^{b-1}(a) < & \{a,b\} & < f_4^b(a) \\ f_3^b(a) < f_3^{f_3^{b-1}(a)-1}(f_3^{b-1}(a)) < & {a,b,2} & < f_4^{f_4^{b-1}(a)}(f_4^{b-1}(a)) < f_5^b(a) \end{eqnarray*}

Therefore your evaluation of \(\{a,b,2\}\) seems incorrect. It does not have the growth rate \(\omega^2\) on \(a\). I am sorry if I am wrong.