User blog comment:Planterobloon/How do you compare huge numbers to each other?/@comment-11227630-20170812012304/@comment-11227630-20170813015657

Your idea works on some "notational" functions, but may fail on some "combinatorial" functions which have some "cost".

For example, Goodstein function has tetrational cost: Then G(100) is much smaller than $$f_{\varepsilon_0}(100)$$.
 * $$G(2)=f_1(3)-3=3$$
 * $$G(4)=f_\omega(3)-3$$
 * $$G(16)=f_{\omega^\omega}(3)-3$$
 * $$G(65536)=f_{\omega^{\omega^\omega}}(3)-3$$
 * $$G(2^{65536})=f_{\omega^{\omega^{\omega^\omega}}}(3)-3$$