The "cheta" function

[jaydfox]

For comparison let me reput your formula from an early thread.
I am only focus on the superfunction not so much on the specific constants such that it is \( 0\mapsto 1 \). Your double limit can be split into two limits, first:

\( T_b(x) = x + b^{x-1+b^{x-2 + b^{\dots}}} \) and then
the superexponential to base b:
\( \operatorname{sexp}_b(x) = \lim_{n\to \infty} \log_b^{[n]} ( T_b(x_0+x+n) ) \)
for a suitable \( x_0 \)

Ah, I remember that approach... I stopped pursuing this particular approach to tetration, because its answers varied from those of the cheta + base-change approach and from Andrew's solution (the "intuitive" solution?). (Not saying it's not at least partially relevant here, but just pointing out that this was my first attempt at a solution, and I dropped it in favor of cheta with base-change, which was my second attempt.)