If $x$ is a real number greater than $e^{e^{-1}}$ , then $x\uparrow \uparrow n$ (A power tower of $n$ $x's$) tends to $\infty$, if $n$ tends to $\infty$. Therefore, there must be a number $n$, such that $x\uparrow\uparrow n>10^{100}$
Can I determine the smallest number $n$ satisfying this inequality without applying the iteration $x_1=x$ , $x_{n+1}=x^{x_n}$ ?
For example, for $\color\red {x=e^{e^{-1}}+10^{-10}}$, we have $\color\green {n=323\ 892}$