User blog comment:Simplicityaboveall/Insanely Fast-Growing Functions & Large Numbers - A Simple Construction/@comment-35470197-20181212123413/@comment-35470197-20181216121322

> f_{k}(1) = 10 for all values of k !!

> f_{k}(1) = f^1_{k-1}(1)

> This is an obvious implication of equation 1.

No. Equation 1 is applicable to \(f_k(n)\) with \(k,n \in \mathbb{N}\) only when \(k < n\). In order to explain the reason, I suggested you twice to write down what values you substituted for \(k\) and \(n\) in equation 1, but you have never follow my advise. Once you had done so, you would understand that \(t+1\) can never be the value of \(lambda_1^t(k)\) for a natural number \(k\) because \(\lambda_1^t(k)\) is ill-defined. Then I understood that you just do not want to understand the problem.

The whole explanation given by you ensures that the one who could not understand what you actually wrote is you. Are you considering a "positional notation in base 1", which is obviously ill-defined?

Anyway, even if you replace the definition so that \(f_{k}(1) = 10\) for any \(k\), the resulting function has growth rate \(< \varepsilon_0\) with respect to FGH through Wainer hierarchy, because the alternative equation 1 gives a system of fundamental sequences on the notation system associated to the canonical well-ordering on trees which is weaker than the Wainer hierarchy. Therefore it is not insanely fast growing.