User blog comment:B1mb0w/Finite: a tour of the finite numbers/@comment-240F:1B:101F:1:E8A0:B3C3:70B1:A854-20170727123133/@comment-28606698-20170727213309

Accurately saing, $$f_2^2(8)=6.61852284341 \times 10^{619}$$

Some remarks for your video


 * 1) $$\omega$$ is not first uncountable number, it  is the first transfinite ordinal. FGH always indexed by a countable ordinals  and defined up to some large countable ordinal $$\mu$$, such that fundamental sequence  is assigned to every limit ordinal less than $$\mu$$.
 * 2) $$\varepsilon_0$$ is not first limit ordinal after $$\omega$$, for example $$\omega+\omega$$ is also a limit ordinal.
 * 3) $$\varepsilon_1$$ is not first limit ordinal after $$\varepsilon_0$$, for example $$\varepsilon_0+\omega$$ is also a limit ordinal.
 * 4) $$\varepsilon_2$$ is not first limit ordinal after $$\varepsilon_1$$
 * 5) Is $$f_{\varepsilon_0}(3)$$ equal to $$f_{\omega^{\omega^\omega}}(3)$$ or to $$f_{\omega^{\omega}}(3)$$ depends on assignation of the fundamental sequences
 * 6) The statement $$\varphi(1,1)=\varepsilon_1=\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}$$ is not correct - this ordinal itself corresponds to infinite tower of epsilons, but you can write $$\varphi(1,1)[3]=\varepsilon_1[3]=\omega^{\omega^{\omega^{\varepsilon_0+1}}}=\varepsilon_0^{\varepsilon_0^\omega}$$
 * 7) On my site you can find a lot of examples for FGH up to epsilon on the page "Fast Growing hierarchy - main road to the infinity"