Euler number

In our context, one can characterize the Euler number e in various ways:

1. As limit of infinite tetration: $e=e_3=\max(b\in\R:b[4]\def\e{\infty}\e\in\R)[4]\e$ This b is called the critical base $\def\ {\eta}\ $. It has the value $\ =\ _3=e^{e^{-1}}\approx1.4446678610097661336583391085964$. The function $\ _x$ is not to be confused with the function $\ph_3(x)$ in Veblen function, which is also denoted $\ _x$.
2. As maximum of the self-root $\sqrt[x]x$: e is the argument x, where $\sqrt[x]x$ has its maximum. The maximum is equal to $\ $.
3. The critical base $\ $ is the base b where the function b^x changes from having 2 fixpoints to having no fixpoint (on the real axis). e is the fixpoint of $\ ^x$.

Now, tetra-Euler and the tetra-critical base is the analogon with using tetration instead of exponentiation (it remains open which extension of tetration we use, and the numeric values may dependently differ, but if not saying differently the candidates are regular tetration for $b\le\ $ and disturbed Fatou coordinates for $b>\ $, which is equal to the Kneser construction).

The tetra-critical base $\ _4\approx1.6353244967152763993453446183062$.
Tetra-Euler $e_4\approx3.0885322718067176544821807826411$.
(Values computed by sheldonison.)

It seems we have the same 3 ways of characterization:

1. Limit of infinite tetration: $\ _4=\max(b\in\R:b[5]\e\in\R)\:e_4=\lim_{n\to\e}b[5]n$ Andrew/Nuninho pointed it out in this thread.
2. Maximum of self-tetra-root: The tetraroot is defined as an inverse of tetration. I.e. $y[/4]x=b\iff b[4]x=y$. The self-tetra-root is then defined as x [/4] x. It has its maximum at $x=e_4$ $\ _4=e_4[/4]e_4$. Mike found out in this thread.
3. The $\ _4$ is the base where b[4]x changes from having 3 fixpoints to having one fixpoint. $e_4$ is the fixpoint into which the 2 fixpoints transform before they vanish. Sheldon's posting contains the following graph and precise values.

A corresponding question/problem about the limit of the Eulers and Etas was asked here.