This is just me approximating the Fast-Growing Hierarchy with Knuth's Arrow Notation and Bird's Array Notation. This is very rough estimate, and it is onlly for me trying to understand BAN.
\(f_0(n)=n+1\)
\(f_1(n)=2n\)
\(f_2(n)=2^nn>2\uparrow n\)
\(f_3(n)>2\uparrow\uparrow n\)
\(f_4(n)>2\uparrow\uparrow\uparrow n\)
\(f_m(n)>2\uparrow^{m-1}n\)
\(f_\omega(n)>2\uparrow^{n-1}n\)
\(f_\omega(n)>2\uparrow^{n-1}n=\{2,n,n-1\}\)
\(f_{\omega+1}(n)\approx\{2,n,1,2\}\)
\(f_{\omega+2}(n)\approx\{2,n,2,2\}\)
\(f_{\omega+3}(n)\approx\{2,n,3,2\}\)
\(f_{\omega+m}(n)\approx\{2,n,m,2\}\)
\(f_{\omega2}(n)\approx\{2,n,n,2\}\)
\(f_{\omega2+1}(n)\approx\{2,n,1,3\}\)
\(f_{\omega2+2}(n)\approx\{2.n.2.3\}\)
\(f_{\omega2+m}(n)\approx\{2,n,m,3\}\)
\(f_{\omega3}(n)\approx\{2,n,n,3\}\)
\(f_{\omega m}(n)\approx\{2,n,n,m\}\)
\(f_{\omega^2}(n)\approx\{2,n,n,n\}\)
\(f_{\omega^2}(n)\approx\{2,n,n,n\}\)
\(f_{\omega^2+1}(n)\approx\{2,n,1,1,2\}\)
\(f_{\omega^2+m}(n)\approx\{2,n,m,1,2\}\)
\(f_{\omega^2+\omega}(n)\approx\{2,n,n,1,2\}\)
\(f_{\omega^2+\omega2}(n)\approx\{2,n,n,2,2\}\)
\(f_{\omega^2+\omega m}(n)\approx\{2,n,n,m,2\}\)
\(f_{\omega^22}(n)\approx\{2,n,n,n,2\}\)
\(f_{\omega^23}(n)\approx\{2,n,n,n,3\}\)
\(f_{\omega^2m}(n)\approx\{2,n,n,n,m\}\)
\(f_{\omega^3}(n)\approx\{2,n,n,n,n\}\)
\(f_{\omega^3+1}(n)\approx\{2,n,1,1,1,2\}\)
\(f_{\omega^3+\omega}(n)\approx\{2,n,n,1,1,2\}\)
\(f_{\omega^3+\omega^2}(n)\approx\{2,n.n,n,1,2\}\)
\(f_{\omega^32}(n)\approx\{2,n,n,n,n,2\}\)
\(f_{\omega^3m}(n)\approx\{2,n,n,n,n,m\}\)
\(\begin{align}f_{\omega^4}(n)&\approx\{2,n,n,n,n,n\}\\&\leq\{n,n,n,n,n,n\}\\&=\{n\langle1\rangle6\}\\&=\{n,6[2]2\}\end{align}\)
\(f_{\omega^m}(n)\approx\{n,m+2[2]2\}\)
\(f_{\omega^\omega}(n)\approx\{n,n+2[2]2\}\)
\(f_{\omega^\omega}(n)\approx\{n,n+2[2]2\}\)
\(f_{\omega^\omega+1}(n)\approx\{n,n,2[2]2\}\)
\(f_{\omega^\omega+\omega}(n)\approx\{n,n,n+1[2]2\}\)
\(f_{\omega^\omega+\omega^2}(n)\approx\{n,n,n,n[2]2\}=\{n,4[2]3\}\)
\(f_{\omega^\omega+\omega^m}(n)\approx\{n,m+2[2]3\}\)
\(f_{\omega^\omega2}(n)\approx\{n,n+2[2]3\}\)
\(f_{\omega^\omega m}(n)\approx\{n,n+2[2]m+1\}\)
\(f_{\omega^{\omega+1}}(n)\approx\{n,n+2[2]n+1\}\)
\(f_{\omega^{\omega+1}+1}(n)\approx\{n,n[2]1,2\}\)
\(f_{\omega^{\omega+1}+\omega^\omega}(n)\approx\{n,n+2[2]2,2\}\)
\(f_{\omega^{\omega+1}2}(n)\approx\{n,n+2[2]n+1,2\}\)
\(f_{\omega^{\omega+1}m}(n)\approx\{n,n+2[2]n+1,m\}\)
\(f_{\omega^{\omega+2}}(n)\approx\{n,n+2[2]n+1,n\}\)
\(f_{\omega^{\omega+2}+1}(n)\approx\{n,n[2]1,1,2\}\)
\(f_{\omega^{\omega+2}m}(n)\approx\{n,n+2[2]n+1,n,m\}\)
\(f_{\omega^{\omega+3}}(n)\approx\{n,n+2[2]n+1,n,n\}\)
\(\begin{align}f_{\omega^{\omega+4}}(n)&\approx\{n,n+2[2]n+1,n,n,n\}\\&\gt\{n,n+2[2]n,n,n,n\}\\&=\{n,n+1[2]n\langle1\rangle4\}\\&\approx\{n,4[2]1[2]2\}\end{align}\)
\(f_{\omega^{\omega+m}}(n)\approx\{n,m[2]1[2]2\}\)
\(f_{\omega^{\omega2}}(n)\approx\{n,n[2]1[2]2\}\)
\(f_{\omega^{\omega2}}(n)\approx\{n,n[2]1[2]2\}\)
\(f_{\omega^{\omega2}+1}(n)\approx\{n,n,2[2]1[2]2\}\)
\(f_{\omega^{\omega2}+\omega^\omega}(n)\approx\{n,n+2[2]2[2]2\}\)
\(f_{\omega^{\omega2}+\omega^{\omega+m}}(n)\approx\{n,m[2]1[2]3\}\)
\(f_{\omega^{\omega2}2}(n)\approx\{n,n[2]1[2]3\}\)
\(f_{\omega^{\omega2}m}(n)\approx\{n,n[2]1[2]m+1\}\)
\(f_{\omega^{\omega2+1}}(n)\approx\{n,n[2]1[2]n+1\}\)
\(f_{\omega^{\omega2+1}+1}(n)\approx\{n,n[2]1[2]1,2\}\)
\(f_{\omega^{\omega2+1}+\omega^{\omega2}m}(n)\approx\{n,n[2]1[2]m+1,2\}\)
\(f_{\omega^{\omega2+1}2}(n)\approx\{n,n[2]1[2]n+1,2\}\)
\(f_{\omega^{\omega2+1}m}(n)\approx\{n,n[2]1[2]n+1,m\}\)
\(f_{\omega^{\omega2+2}}(n)\approx\{n,n[2]1[2]n+1,n\}\)
\(f_{\omega^{\omega2+2}2}(n)\approx\{n,n[2]1[2]n+1,n,2\}\)
\(f_{\omega^{\omega2+2}m}(n)\approx\{n,n[2]1[2]n+1,n,m\}\)
\(f_{\omega^{\omega2+3}}(n)\approx\{n,n[2]1[2]n+1,n,n\}\)
\(\begin{align}f_{\omega^{\omega2+4}}(n)&\approx\{n,n[2]1[2]n+1,n,n,n\}\\&\gt\{n,n[2]1[2]n\langle1\rangle4\}\\&\approx\{n,4[2]1[2]1[2]2\}\end{align}\)
\(f_{\omega^{\omega3}}(n)\approx\{n,n[2]1[2]1[2]2\}\)
\(\begin{align}f_{\omega^{\omega4}}(n)&\approx\{n,n[2]1[2]1[2]1[2]2\}\\&\le\{n\langle2\rangle5\}\\&=\{n,5[3]2\}\end{align}\)
\(f_{\omega^{\omega m}}(n)\approx\{n,m+1[3]2\}\)
\(f_{\omega^{\omega^2}}(n)\approx\{n,n+1[3]2\}\)
\(f_{\omega^{\omega^2}}(n)\approx\{n,n+1[3]2\}\)
\(f_{\omega^{\omega^2}+1}(n)\approx\{n,n,2[3]2\}\)
\(f_{\omega^{\omega^2}+\omega^\omega}(n)\approx\{n,n+2[2]2[3]2\}\)
\(f_{\omega^{\omega^2}+\omega^{\omega2}}(n)\approx\{n,n[2]1[2]2[3]2\}\)
\(f_{\omega^{\omega^2}2}(n)\approx\{n,n+1[3]3\}\)
\(f_{\omega^{\omega^2+1}}(n)\approx\{n,n+1[3]n+1\}\)
\(f_{\omega^{\omega^2+2}}(n)\approx\{n,n+1[3]n+1,n\}\)
\(f_{\omega^{\omega^2+\omega}}(n)\approx\{n,n[3]1[2]2\}\)
\(f_{\omega^{\omega^2+\omega}2}(n)\approx\{n,n[3]1[2]3\}\)
\(f_{\omega^{\omega^2+\omega+1}}(n)\approx\{n,n[3]1[2]n+1\}\)
\(f_{\omega^{\omega^2+\omega+2}}(n)\approx\{n,n[3]1[2]n+1,n\}\)
\(f_{\omega^{\omega^2+\omega2}}(n)\approx\{n,n[3]1[2]1[2]2\}\)
\(f_{\omega^{\omega^2+\omega m}}(n)\approx\{n,m+1[3]1[3]2\}\)
\(f_{\omega^{\omega^22}}(n)\approx\{n,n+1[3]1[3]2\}\)
\(f_{\omega^{\omega^23}}(n)\approx\{n,n+1[3]1[3]1[3]2\}\)
\(f_{\omega^{\omega^3}}(n)\approx\{n,n+1[4]2\}\)
\(f_{\omega^{\omega^4}}(n)\approx\{n,n+1[5]2\}\)
\(f_{\omega^{\omega^m}}(n)\approx\{n,n+1[m+1]2\}\)
\(\begin{align}f_{\omega^{\omega^\omega}}(n)&\approx\{n,n+1[n+1]2\}\\&\approx\{n,n[1,2]2\}\end{align}\)
\(f_{\omega^{\omega^\omega}}(n)\approx\{n,n[1,2]2\}\)
\(f_{\omega^{\omega^\omega}+1}(n)\approx\{n,n,2[1,2]2\}\)
\(f_{\omega^{\omega^\omega}2}(n)\approx\{n,n[1,2]3\}\)
\(f_{\omega^{\omega^\omega+1}}(n)\approx\{n,n[1,2]n+1\}\)
\(f_{\omega^{\omega^\omega+2}}(n)\approx\{n,n[1,2]n+1,n\}\)
\(f_{\omega^{\omega^\omega+\omega}}(n)\approx\{n,n[1,2]1[2]2\}\)
\(f_{\omega^{\omega^\omega+\omega^2}}(n)\approx\{n,n[1,2]1[3]2\}\)
\(f_{\omega^{\omega^\omega2}}(n)\approx\{n,n[1,2]1[1,2]2\}\)
\(f_{\omega^{\omega^{\omega+1}}}(n)\approx\{n,n+1[2,2]2\}\)
\(f_{\omega^{\omega^{\omega+1}}2}(n)\approx\{n,n+1[2,2]3\}\)
\(f_{\omega^{\omega^{\omega+1}+1}}(n)\approx\{n,n+1[2,2]n+1\}\)
\(f_{\omega^{\omega^{\omega+1}2}}(n)\approx\{n,n+1[2,2]1[2,2]2\}\)
\(f_{\omega^{\omega^{\omega+2}}}(n)\approx\{n,n+1[3,2],2\}\)
\(\begin{align}f_{\omega^{\omega^{\omega+m}}}(n)&\approx\{n,n[m+1,2]2\}\\&\approx\{n,m+1[1,3]2\}\end{align}\)
\(f_{\omega^{\omega^{\omega2}}}(n)\approx\{n,n+1[1,3]2\}\)
\(\begin{align}f_{\omega^{\omega^{\omega m}}}(n)&\approx\{n,n+1[1,m+1]2\}\\&\approx\{n,m[1,1,2]2\}\end{align}\)
\(f_{\omega^{\omega^{\omega^2}}}(n)\approx\{n,n[1,1,2]2\}\)
\(f_{\omega^{\omega^{\omega^2}}2}(n)\approx\{n,n[1,1,2]3\}\)
\(f_{\omega^{\omega^{\omega^2}2}}(n)\approx\{n,n[1,1,2]1[1,1,2]2\}\)
\(f_{\omega^{\omega^{\omega^22}}}(n)\approx\{n,n[1,1,3]2\}\)
\(\begin{align}f_{\omega^{\omega^{\omega^3}}}(n)&\approx\{n,n[1,1,n+1]2\}\\&\approx\{n,n[1,1,1,2]2\}\end{align}\)
\(\begin{align}f_{\omega^{\omega^{\omega^m}}}(n)&\approx\{n,n[n\langle1\rangle m]2\}\\&\approx\{n,m[1[2]2]2\}\end{align}\)
\(f_{\omega^{\omega^{\omega^\omega}}}(n)\approx\{n,n[1[2]2]2\}\)
\(f_{\omega^{\omega^{\omega^\omega}}}(n)\approx\{n,n[1[2]2]2\}\)
\(f_{\omega^{\omega^{\omega^\omega}+\omega}}(n)\approx\{n,n[1[2]2]1[2]2\}\)
\(f_{\omega^{\omega^{\omega^\omega}+\omega^\omega}}(n)\approx\{n,n[1[2]2]1[1,2]2\}\)
\(f_{\omega^{\omega^{\omega^\omega}2}}(n)\approx\{n,n[1[2]2]1[1[2]2]2\}\)
\(f_{\omega^{\omega^{\omega^\omega+1}}}(n)\approx\{n,n+1[2[2]2]2\}\)
\(f_{\omega^{\omega^{\omega^\omega+\omega}}}(n)\approx\{n,n+1[1,2[2]2]2\}\)
\(f_{\omega^{\omega^{\omega^\omega+\omega2}}}(n)\approx\{n,n+1[1,3[2]2]2\}\)
\(f_{\omega^{\omega^{\omega^\omega+\omega^2}}}(n)\approx\{n,n+1[1,1,2[2]2]2\}\)
\(f_{\omega^{\omega^{\omega^\omega2}}}(n)\approx\{n,n+1[1[2]3]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega+1}}}}(n)\approx\{n,n+1[1[2]1,2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega2}}}}(n)\approx\{n,n+1[1[2]1[2]2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega^2}}}}(n)\approx\{n,n+1[1[3]2]\}\)
\(f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(n)\approx\{n,n+1[1[1,2]2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega^\omega}}}+1}(n)\approx\{n,n,2[1[1,2]2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega^\omega}}+1}}(n)\approx\{n,n+1[1[1,2]2]n+1\}\)
\(f_{\omega^{\omega^{\omega^{\omega^\omega}+1}}}(n)\approx\{n,n+1[2[1,2]2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega^\omega+1}}}}(n)\approx\{n,n+1[1[1,2]1,2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega^\omega+\omega}}}}(n)\approx\{n,n+1[1[1,2]1[2]2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega^\omega2}}}}(n)\approx\{n,n+1[1[1,2]1[1,2]2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega^{\omega+1}}}}}(n)\approx\{n,n+1[1[2,2]2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega^{\omega2}}}}}(n)\approx\{n,n+1[1[1,3]2]2\}\)
\(f_{\omega^{\omega^{\omega^{\omega^{\omega^2}}}}}(n)\approx\{n,n+1[1[1,1,2]2]2\}\)
\(\begin{align}f_{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}(n)&\approx\{n,n+1[1[1[2]2]2]2\}\\&\approx\{n,n[2\langle0\backslash2\rangle5]2\}\\&\approx\{n,5[1\backslash2]2\}\end{align}\)
\(f_{\varepsilon_0}(n)\approx\{n,n[1\backslash2]2\}\)
\(f_{\varepsilon_0}(n)\approx\{n,n[1\backslash2]2\}\)
\(f_{\varepsilon_0+1}(n)\approx\{n,n,2[1\backslash2]2\}\)
\(f_{\varepsilon_0+\omega}(n)\approx\{n,n,n[1\backslash2]2\}\)
\(f_{\varepsilon_0+\omega^2}(n)\approx\{n,n,n,n[1\backslash2]2\}\)
\(f_{\varepsilon_0+\omega^\omega}(n)\approx\{n,n+2[2]2[1\backslash2]2\}\)
\(f_{\varepsilon_0+\omega^{\omega^\omega}}(n)\approx\{n,n[1,2]2[1\backslash2]2\}\)
\(f_{\varepsilon_02}(n)\approx\{n,n[1\backslash2]3\}\)
\(f_{\omega^{\varepsilon_0+1}}(n)\approx\{n,n[1\backslash2]n+1\}\)
\(f_{\omega^{\omega^{\varepsilon_0+1}}}(n)\approx\{n,n[2\backslash2]2\}\)
\(f_{\omega^{\omega^{\omega^{\varepsilon_0+1}}}}(n)\approx\{n,n[1,2\backslash2]2\}\)
\(f_{\varepsilon_1}(n)\approx\{n,n[1\backslash3]2\}\)
\(f_{\varepsilon_2}(n)\approx\{n,n[1\backslash4]2\}\)
\(f_{\varepsilon_\omega}(n)\approx\{n,n+2[1\backslash1,2]2\}\)
\(f_{\varepsilon_\omega}(n)\approx\{n,n+2[1\backslash1,2]2\}\)
\(f_{\varepsilon_{\omega+1}}(n)\approx\{n,n[1\backslash2,2]2\}\)
\(f_{\varepsilon_{\omega2}}(n)\approx\{n,n+1[1\backslash1,3]2\}\)
\(f_{\varepsilon_{\omega^2}}(n)\approx\{n,n+1[1\backslash1,1,2]2\}\)
\(f_{\varepsilon_{\omega^3}}(n)\approx\{n,n+1[1\backslash1,1,1,2]2\}\)
\(f_{\varepsilon_{\omega^\omega}}(n)\approx\{n,n+1[1\backslash1[2]2]2\}\)
\(f_{\varepsilon_{\omega^{\omega2}}}(n)\approx\{n,n+1[1\backslash1[2]1[2]2]2\}\)
\(f_{\varepsilon_{\omega^{\omega^2}}}(n)\approx\{n,n+1[1\backslash1[3]2]2\}\)
\(f_{\varepsilon_{\omega^{\omega^\omega}}}(n)\approx\{n,n+1[1\backslash1[1,2]2]2\}\)
\(f_{\varepsilon_{\omega^{\omega^{\omega^\omega}}}}(n)\approx\{n,n+1[1\backslash1[1[2]2]2]2\}\)
\(f_{\varepsilon_{\varepsilon_0}}(n)\approx\{n,n[1\backslash1[1\backslash2]2]2\}\)
\(f_{\varepsilon_{\varepsilon_{\varepsilon_0}}}(n)\approx\{n,n[1\backslash1[1\backslash1[1\backslash2]2]2]2\}\)
\(f_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}}(n)\approx\{n,n[1\backslash1[1\backslash1[1\backslash1[1\backslash2]2]2]2]2\}\)
\(f_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}}}(n)\approx\{n,n[1\backslash1[1\backslash1[1\backslash1[1\backslash1[1\backslash2]2]2]2]2]2\}\)
\(f_{\zeta_0}(n)\approx\{n,n[1\backslash1\backslash2]2\}\)
\(f_{\zeta_0}(n)\approx\{n,n[1\backslash1\backslash2]2\}\)
\(f_{\zeta_0+1}(n)\approx\{n,n,2[1\backslash1\backslash2]2\}\)
\(f_{\zeta_0+\omega}(n)\approx\{n,n,n+1[1\backslash1\backslash2]2\}\)
\(f_{\zeta_0+\varepsilon_0}(n)\approx\{n,n[1\backslash2]2[1\backslash1\backslash2]2\}\)
\(f_{\zeta_0+\varepsilon_{\varepsilon_0}}(n)\approx\{n,n[1\backslash1[1\backslash2]2]1[1\backslash1\backslash2]2\}\)
\(f_{\zeta_02}(n)\approx\{n,n[1\backslash1\backslash2]3\}\)
\(f_{\omega^{\zeta_0+1}}(n)\approx\{n,n[1\backslash1\backslash2]n+1\}\)
\(f_{\varepsilon_{\zeta_0+1}}(n)\approx\{n,n[1\backslash2\backslash2]2\}\)
\(f_{\varepsilon_{\zeta_0+\omega}}(n)\approx\{n,n[1\backslash1,2\backslash2]2\}\)
\(f_{\varepsilon_{\zeta_0+\omega^\omega}}(n)\approx\{n,n[1\backslash1[2]2\backslash2]2\}\)
\(f_{\varepsilon_{\zeta_0+\varepsilon_0}}(n)\approx\{n,n[1\backslash1[1\backslash2]2\backslash2]2\}\)
\(f_{\varepsilon_{\zeta_02}}(n)\approx\{n,n[1\backslash1[1\backslash1\backslash2]2\backslash2]2\}\)
\(f_{\varepsilon_{\omega^{\zeta_0+1}}}(n)\approx\{n,n[1\backslash1[1\backslash1\backslash2]1,2\backslash2]2\}\)
\(f_{\varepsilon_{\varepsilon_{\zeta_0+1}}}(n)\approx\{n,n[1\backslash1[1\backslash2\backslash2]2\backslash2]2\}\)
\(f_{\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0+1}}}}(n)\approx\{n,n[1\backslash1[1\backslash1[1\backslash1\backslash2]2\backslash2]2\backslash2]2\}\)
\(f_{\zeta_1}(n)\approx\{n,n[1\backslash1\backslash3]2\}\)
\(f_{\zeta_\omega}(n)\approx\{n,n+1[1\backslash1\backslash1,2]2\}\)
\(f_{\zeta_{\omega^\omega}}(n)\approx\{n,n[1\backslash1\backslash1[2]2]2\}\)
\(f_{\zeta_{\omega^{\omega^\omega}}}(n)\approx\{n,n[1\backslash1\backslash1[1,2]2]2\}\)
\(f_{\zeta_{\varepsilon_0}}(n)\approx\{n,n[1\backslash1\backslash1[1\backslash2]2]2\}\)
\(f_{\zeta_{\varepsilon_{\varepsilon_0}}}(n)\approx\{n,n[1\backslash1\backslash1[1\backslash1[1\backslash2]2]2]2\}\)
\(f_{\zeta_{\zeta_0}}(n)\approx\{n,n[1\backslash1\backslash1[1\backslash1\backslash2]2]2\}\)
\(f_{\zeta_{\zeta_{\zeta_0}}}(n)\approx\{n,n[1\backslash1\backslash1[1\backslash1\backslash1[1\backslash1\backslash2]2]2]2\}\)
\(f_{\zeta_{\zeta_{\zeta_{\zeta_0}}}}(n)\approx\{n,n[1\backslash1\backslash1[1\backslash1\backslash1[1\backslash1\backslash1[1\backslash1\backslash2]2]2]2]2\}\)
\(f_{\eta_0}(n)\approx\{n,n[1\backslash1\backslash1\backslash2]2\}\)
\(f_{\eta_0}(n)\approx\{n,n[1\backslash1\backslash1\backslash2]2\}\)
\(f_{\eta_1}(n)\approx\{n,n[1\backslash1\backslash1\backslash3]2\}\)
\(f_{\eta_\omega}(n)\approx\{n,n+1[1\backslash1\backslash1\backslash1,2]2\}\)
\(f_{\eta_{\varepsilon_0}}(n)\approx\{n,n[1\backslash1\backslash1\backslash1[1\backslash2]2]2\}\)
\(f_{\eta_{\zeta_0}}(n)\approx\{n,n[1\backslash1\backslash1\backslash1[1\backslash1\backslash2]2]2\}\)
\(f_{\eta_{\eta_0}}(n)\approx\{n,n[1\backslash1\backslash1\backslash1[1\backslash1\backslash1\backslash2]2]2\}\)
\(f_{\eta_{\eta_{\eta_0}}}(n)\approx\{n,n[1\backslash1\backslash1\backslash1[1\backslash1\backslash1\backslash1[1\backslash1\backslash1\backslash2]2]2]2\}\)
\(f_{\varphi_4(0)}(n)\approx\{n,n[1\backslash1\backslash1\backslash1\backslash2]2\}\)
\(f_{\varphi_4^2(0)}(n)\approx\{n,n[1\backslash1\backslash1\backslash1\backslash1[1\backslash1\backslash1\backslash1\backslash2]2]2\}\)
\(f_{\varphi_5(0)}(n)\approx\{n,n[1\backslash1\backslash1\backslash1\backslash1\backslash2]2\}\)
\(f_{\varphi_m(0)}(n)\approx\{n,n[2\langle1\rangle\backslash m+1]2\}\)
\(f_{\varphi_\omega(0)}(n)\approx\{n,n+1[1[2]\backslash2]2\}\)
\(f_{\varphi_\omega(0)}(n)\approx\{n,n+1[1[2]\backslash2]2\}\)
\(f_{\varphi_{\omega+1}(0)}(n)\approx\{n,n[1[2]\backslash1\backslash2]2\}\)
\(f_{\varphi_{\omega2}(0)}(n)\approx\{n,n[1[2]\backslash1[2]\backslash2]2\}\)
\(f_{\varphi_{\omega^2}(0)}(n)\approx\{n,n[1[3]\backslash2]2\}\)
\(f_{\varphi_{\omega^3}(0)}(n)\approx\{n,n[1[4]\backslash2]2\}\)
\(f_{\varphi_{\omega^\omega}(0)}(n)\approx\{n,n[1[1,2]\backslash2]2\}\)