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Well I had to do it at some point.

Linear arrays - up to \(\omega^\omega\)

\(b(a,a)\)

\(a^2\)
\(b(a,a,1)\) \(f_2(a)\)
\(b(a,a,n)\) \(f_{n+1}(a)\)
\(b(a,a,0,1)\) \(f_{\omega}(a)\)
\(b(a,a,n,1)\) \(f_{\omega +n}(a)\)
\(b(a,a,n,k)\) \(f_{\omega k + n}(a)\)
\(b(a,a,0,0,1)\) \(f_{\omega^2}(a)\)
\(b(a,a,n,k,i)\) \(f_{\omega^2 i + \omega k + n}(a)\)
\(b(a,a,0,0,0,1)\) \(f_{\omega^3}(a)\)
\(b(a,a,n,k,l,m)\) \(f_{\omega^3 m + \omega^2 l + \omega k + n}(a)\)
\(b(a,a,b,c,d,e,.....)\) \(f_{...... + \omega^3 e + \omega^2 d + \omega c +b}(a)\)

Limit is \(\omega^\omega\).

Two rows - up to \(\omega^{\omega 2}\)

\(b(a,a\{1\}1)\) \(f_{\omega^\omega}(a)\)
\(b(a,a,1\{1\}1)\) \(f_{\omega^\omega +1}(a)\)
\(b(a,a,n\{1\}1)\) \(f_{\omega^\omega +n}(a)\)
\(b(a,a,0,1\{1\}1)\) \(f_{\omega^\omega + \omega}(a)\)
\(b(a,a,n,k\{1\}1)\) \(f_{\omega^\omega + \omega k + n}(a)\)
\(b(a,a,b,c,d,.....\{1\}1)\) \(f_{\omega^\omega + ..... +\omega^2 d + \omega c + b}(a)\)
\(b(a,a\{1\}2)\) \(f_{\omega^\omega 2}(a)\)
\(b(a,a,b,c,d,e,....\{1\}n)\)                                  \(f_{\omega^\omega n + ...... \omega^3 e + \omega^2 d + \omega c +b}(a)\)
\(b(a,a\{1\}0,1)\) \(f_{\omega^{\omega +1}}(a)\)
\(b(a,a,b,c,d,e,...\{1\}n,1)\)                    \(f_{\omega^{\omega +1} + \omega^\omega n + ...... \omega^3 e + \omega^2 d + \omega c +b}(a)\)
\(b(a,a\{1\}0,2)\) \(f_{\omega^{\omega +1} 2}(a)\)
\(b(a,a\{1\}0,0,1)\) \(f_{\omega^{\omega +2}}(a)\)
\(b(a,a\{1\}n_1,n_2,.....,n_i,n_{i+1})\) \(f_{\omega^{\omega + i}n_{i+1} + \omega^{\omega + i - 1}n_{i} + ...... + \omega^{\omega +1}n_2 + \omega^{\omega}n_1}(a)\)

Planar - up to \(\omega^{\omega^2}\)

\(b(a,a\{1\}0\{1\}1)\) \(f_{\omega^{\omega 2}}(a)\)
\(b(a,a,1\{1\}0\{1\}1)\) \(f_{\omega^{\omega 2} +1}(a)\)
\(b(a,a,b,c,d,e....\{1\}0\{1\}1)\) \(f_{\omega^{\omega 2} +.....+\omega^3 e \omega^2 d + \omega c + b}(a)\)
\(b(a,a\{1\}1\{1\}1)\) \(f_{\omega^{\omega 2} + \omega^\omega}(a)\)
\(b(a,a\{1\}b,c,d,e,....\{1\}1)\)                          \(f_{\omega^{\omega 2} + ..... + \omega^{\omega +3}e +\omega^{\omega +2}d + \omega^{\omega +1}c + \omega^\omega b}\)
\(b(a,a\{1\}0\{1\}2)\) \(f_{\omega^{\omega 2} 2}(a)\)
\(b(a,a\{1\}0\{1\}b,c,d,e,...)\) \(f_{..... + \omega^{\omega 2 + 3}e + \omega^{\omega 2 + 2}d + \omega^{\omega 2 + 1}c + \omega^{\omega 2}b}\)
\(b(a,a\{1\}0\{1\}0\{1\}1)\) \(f_{\omega^{\omega 3}}(a)\)
\(b(a,a\{1\}n_1\{1\}n_2,......\{1\}n_i)\) \(f_{\omega^{\omega i}n_i + ..... + \omega^{\omega 2}n_2 + \omega^{\omega}n_1}(a)\)
\(b(a,a\{1\}0\{1\}0\{1\}0\{1\}.....\{1\}0\{1\}1)\) w/ \(n\) \(\{1\}\)'s                  

\(f_{\omega^{\omega n}}(a)\)

Cubic - up to \(\omega^{\omega^3}\)

\(b(a,a\{2\}1)\)

\(f_{\omega^{\omega^2}}(a)\)
\(b(a,a,1\{2\}1)\) \(f_{\omega^{\omega^2}+1}(a)\)
\(b(a,a,0,1\{2\}1)\) \(f_{\omega^{\omega^2}+\omega}(a)\)
\(b(a,a\{1\}0\{2\}1)\) \(f_{\omega^{\omega^2} + \omega^\omega}(a)\)
\(b(a,a\{1\}1\{2\}1)\) \(f_{\omega^{\omega^2}+\omega^\omega 2}(a)\)
\(b(a,a\{1\}0,1\{2\}1)\) \(f_{\omega^{\omega^2}+\omega^{\omega +1}}(a)\)
\(b(a,a\{1\}0\{1\}0\{2\}1)\) \(f_{\omega^{\omega^2}+\omega^{\omega 2}}(a)\)
\(b(a,a\{2\}2)\) \(f_{\omega^{\omega^2}2}(a)\)
\(b(a,a\{2\}0,1)\) \(f_{\omega^{\omega^2 +1}}(a)\)
\(b(a,a\{2\}0\{1\}1)\) \(f_{\omega^{\omega^2 + \omega}}(a)\)
\(b(a,a\{2\}0\{1\}0\{1\}1)\) \(f_{\omega^{\omega^2 + \omega 2}}\)
\(b(a,a\{2\}0\{2\}1)\) \(f_{\omega^{\omega^2 2}}(a)\)
\(b(a,a\{2\}0\{2\}0\{2\}1)\) \(f_{\omega^{\omega^2 3}}(a)\)

\(b(a,a\{2\}0\{2\}0\{2\}.....\{2\}1)\) w/ \(n\) \(\{2\}\)'s

\(f_{\omega^{\omega^2 n}}(a)\)

Multi-dimentional - up to \(\omega^{\omega^\omega}\)

\(b(a,a\{3\}1)\) \(f_{\omega^{\omega^3}}(a)\)
\(b(a,a,1\{3\}1)\) \(f_{\omega^{\omega^3}+1}(a)\)
\(b(a,a,0,1\{3\}1)\) \(f_{\omega^{\omega^3}+\omega}(a)\)
\(b(a,a\{1\}0\{3\}1)\) \(f_{\omega^{\omega^3}+\omega^\omega}(a)\)
\(b(a,a\{2\}0\{3\}1)\) \(f_{\omega^{\omega^3}+\omega^{\omega^2}}(a)\)
\(b(a,a\{3\}2)\) \(f_{\omega^{\omega^3}2}(a)\)
\(b(a,a\{3\}0,1)\) \(f_{\omega^{\omega^3 +1}}\)
\(b(a,a\{3\}0\{1\}1)\) \(f_{\omega^{\omega^3 +\omega}}(a)\)
\(b(a,a\{3\}0\{3\}1)\) \(f_{\omega^{\omega^3 2}}(a)\)
\(b(a,a\{4\}1)\) \(f_{\omega^{\omega^4}}(a)\)
\(b(a,a\{4\}0,1)\) \(f_{\omega^{\omega^4 +1}}(a)\)
\(b(a,a\{4\}0\{4\}1)\) \(f_{\omega^{\omega^4 2}}(a)\)
\(b(a,a\{4\}0\{4\}0\{4\}......\{4\}1)\) w/ \(n\) \(\{4\}\)'s \(f_{\omega^{\omega^4 n}}(a)\)
\(b(a,a{5}1)\) \(f_{\omega^{\omega^5}}(a)\)
\(b(a,a\{n\}1)\) \(f_{\omega^{\omega^n}}(a)\)

Limit of this is \(\omega^{\omega^\omega}\)

Hyperdimentional - up to \(\varepsilon_0\)

\(b(a,a\{0,1\}1)\) \(f_{\omega^{\omega^\omega}}(a)\)
\(b(a,a,1\{0,1\}1)\) \(f_{\omega^{\omega^\omega}+1}(a)\)
\(b(a,a\{1\}0\{0,1\}1)\) \(f_{\omega^{\omega^\omega}+\omega^\omega}(a)\)
\(b(a,a\{n\}0\{0,1\}1)\) \(f_{\omega^{\omega^\omega}+\omega^{\omega^n}}(a)\)
\(b(a,a\{0,1\}2)\) \(f_{\omega^{\omega^\omega}2}(a)\)
\(b(a,a\{0,1\}0,1)\) \(f_{\omega^{\omega^\omega +1}}(a)\)
\(b(a,a\{0,1\}0\{1\}1)\) \(f_{\omega^{\omega^\omega +\omega}}(a)\)
\(b(a,a\{0,1\}0\{0,1\}1)\) \(f_{\omega^{\omega^\omega 2}}(a)\)
\(b(a,a\{1,1\}1)\) \(f_{\omega^{\omega^{\omega +1}}}(a)\)
\(b(a,a\{n,1\}1)\) \(f_{\omega^{\omega^{\omega +n}}}(a)\)
\(b(a,a\{0,2\}1)\) \(f_{\omega^{\omega^{\omega 2}}}(a)\)
\(b(a,a\{b,c,d,e,....\}1)\) \(f_{\omega^{\omega^{.... +\omega^3 e + \omega^2 d + \omega c +b}}}(a)\)
\(b(a,a\{0\{1\}1\}1)\) \(f_{\omega^{\omega^{\omega^\omega}}}(a)\)
\(b(a,a\{0\{n\}1\}1)\) \(f_{\omega^{\omega^{\omega^{\omega^n}}}}(a)\)
\(b(a,a\{0\{b,c,d,e,...\}1\}1)\) \(f_{\omega^{\omega^{\omega^{\omega^{.....+\omega^3 e + \omega^2 d + \omega c +b}}}}}(a)\)
\(b(a,a\{0\{0\{1\}1\}1\}1)\) \(f_{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}(a)\)
\(b(a,a\{0\{0\{b,c,d,e,...\}1\}1\}1)\) \(f_{\omega^{\omega^{\omega^{\omega^{\omega^{.....+\omega^3 e + \omega^2 d + \omega c +b}}}}}}(a)\)
\(b(a,a\{0\{0\{0\{1\}1\}1\}1\}1)\) \(f_{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}}(a)\)
\(b(a,a\{0\{0\{0\{0,1\}1\}1\}1\}1)\) \(f_{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}}}(a)\)
\(b(a,a\{0\{0\{0\{..\{1\}..\}1\}1\}1\}1)\) w/ \(n\) ones from the center out.       \(f_{\omega\uparrow\uparrow (n*2)}(a)\)

Limit is \(\varepsilon_0\).

Nested - up to \(\varphi(\omega,0)\)

Up to \(\varepsilon_1\)

\(b(a,a{0\backslash 1}1)\) \(f_{\varepsilon_0}(a)\)
\(b(a,a,1\{0\backslash 1\}1)\) \(f_{\varepsilon_0 +1}(a)\)
\(b(a,a,0,1\{0\backslash 1\}1)\) \(f_{\varepsilon_0 +\omega}(a)\)
\(b(a,a\{1\}0\{0\backslash 1\}1)\) \(f_{\varepsilon_0 +\omega^\omega}(a)\)
\(b(a,a\{0,1\}0\{0\backslash 1\}1)\) \(f_{\varepsilon_0 +\omega^{\omega^\omega}}(a)\)
\(b(a,a\{0\{1\}1\}0\{0\backslash 1\}1)\) \(f_{\varepsilon_0 +\omega^{\omega^{\omega^\omega}}}(a)\)
\(b(a,a\{0\{0,1\}1\}0\{0\backslash 1\}1)\) \(f_{\varepsilon_0 +\omega^{\omega^{\omega^{\omega^\omega}}}}(a)\)
\(b(a,a\{0\backslash 1\}2)\) \(f_{\varepsilon_0 2}(a)\)
\(b(a,a\{0\backslash 1\}0,1)\) \(f_{\varepsilon_0 \omega}(a)\)
\(b(a,a\{0\backslash 1\}0\{1\}1)\) \(f_{\varepsilon_0 \omega^\omega}(a)\)
\(b(a,a\{0\backslash 1\}0\{0,1\}1)\) \(f_{\varepsilon_0 \omega^{\omega^\omega}}(a)\)
\(b(a,a\{0\backslash 1\}0\{0\backslash 1\}1)\) \(f_{\varepsilon_0^2}(a)\)
\(b(a,a\{0\backslash 1\}0\{0\backslash 1\}0\{0\backslash 1\}1)\) \(f_{\varepsilon_0^3}(a)\)
\(b(a,a\{1\backslash 1\}1)\) \(f_{\varepsilon_0^\omega}(a)\)
\(b(a,a\{1\backslash 1\}n)\) \(f_{\varepsilon_0^\omega n}(a)\)
\(b(a,a\{1\backslash 1\}0\{1\backslash 1\}1)\) \(f_{\varepsilon_0^{\omega 2}}(a)\)
\(b(a,a\{2\backslash 1\}1)\) \(f_{\varepsilon_0^{\omega^2}}(a)\)
\(b(a,a\{n\backslash 1\}1)\) \(f_{\varepsilon_0^{\omega^n}}(a)\)
\(b(a,a\{0,1\backslash 1\}1)\) \(f_{\varepsilon_0^{\omega^\omega}}(a)\)
\(b(a,a\{0\{1\}1\backslash 1\}1)\) \(f_{\varepsilon_0^{\omega^{\omega^\omega}}}(a)\)
\(b(a,a\{0\{0,1\}1\backslash 1\}1)\) \(f_{\varepsilon_0^{\omega^{\omega^{\omega^\omega}}}}(a)\)
\(b(a,a\{0\{0\backslash1\}1\backslash 1\}1)\) \(f_{\varepsilon_0^{\varepsilon_0}}(a)\)
\(b(a,a\{0\{0\{0\backslash 1\}1\backslash 1\}1\backslash 1\}1)\) \(f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}(a)\)
\(b(a,a\{0\{0\{0\{0\backslash 1\}1\backslash 1\}1\backslash 1\}1\backslash 1\}1)\) \(f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}}(a)\)
\(b(a,a\{0\backslash 2\}1)\) \(f_{\varepsilon_1}(a)\)

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