User blog comment:Primussupremus/Does any one know if there is a really good tutorial on the fast growing hierarchy./@comment-28606698-20170309103653

There are very simple rules:
 * \(f_0(n) = n + 1\)
 * \(f_{\alpha+1}(n) = f^n_\alpha(n)\), where \(f^n\) denotes function iteration
 * \(f_\alpha(n) = f_{\alpha[n]}(n)\) if and only if \(\alpha\) is a limit ordinal

For example: \(f_0(4) = 4 + 1=5\)

\(f_{1}(4) = f^4_0(4)=f_0(f_0(f_0(f_0(4))))=(((4+1)+1)+1)+1=8=4*2\),

\(f_{2}(4) = f^4_1(4)=f_1(f_1(f_1(f_1(4))))=64=(((4*2)*2)*2)*2=4*2^4\),

\(f_{3}(4) = f^4_2(4)=f_2(f_2(f_2(f_2(4))))=\) \(=((4*2^4)*2^(4*2^4))*2^((4*2^4)*2^(4*2^4))*2^((4*2^4)*2^(4*2^4))*2^((4*2^4)*2^(4*2^4))=\)

\(=10^{10^{3.553934904655 * 10^{20}}}\),

\(f_\omega(4) = f_{\omega[4]}(4)\) = f_{4}(4)= f^4_3(4)=f_3(f_3(f_3(f_3(4))))\)

and so on.