User blog comment:P進大好きbot/Elementary Large Number/@comment-31966679-20181117164807/@comment-35470197-20181117223526

If I am correct, your \(F(n)\) is just \(n^n\), which is much smaller than \(f_{\varepsilon_0}(n)\). \begin{eqnarray*} n^n & \ll & f_3(n) \ll f_{\omega}(n) \ll f_{\omega^2}(n) \ll f_{\omega^{\omega}}(n) \\ & \ll & f_{\omega^{\omega^{\omega}}}(n) \ll f_{\varepsilon_0}(n) \ll \textrm{AC}_1(n) \end{eqnarray*}