User blog comment:GamesFan2000/Hyper Tower Notation, Pt. 1 (Inspired by Hyper-E notation)/@comment-35470197-20190318151059/@comment-35470197-20190319010708

I asked it because I could not understand your "analysis". I do not think that your notation is so strong as you stated.

For positive variables, we have the following estimation: \begin{eqnarray*} n_0 \# n_1 & \ll & f_2^{2n_1}(n_0) \\ n_0 \# n_1 \# n_2 & \ll & f_3^{2n_2}(f_2^{2n_1}(n_0)) \\ n_0 \# n_1 \# \cdots \# n_k & \ll & f_{k+1}^{2n_k}(\cdots(f_2^{2n_1}(n_0)) \cdots) \\ n_0 \#_2 n_1 & \ll & f_{n_0}^{2(n_1-1)}(f_{n_0+1}(2n_0+1)) < f_{n_0}^{2n_1}(f_{n_0+1}(2n_0+1)) \\ n \#_2 n & \ll & f_n^{4n-1}(2n+1) \leq f_{n+1}(4n-1) < f_{\omega}(4n-1) \\ n_0 \#_2 n_1 \# n_2 & \ll & f_{n_0}^{2n_1+2n_2}(f_{n_0+1}(2n_0+1)) \\ n \#_2 n \#_2 n & \ll & f_n^{6n+1}(2n+1) < f_{n+1}(6n+1) \ll f_{\omega+1}(n) \end{eqnarray*} It obviously conflicts your description. Maybe I mistook something. Please point out what is wrong.