I know this isn't what you mean but:
\[
f(x) = \sum_{n=0}^\infty \frac{x^n}{2^{2^{2^n}}}
\]
Has triple exponential convergence!
I'm not sure we're there yet. I think with ramanujan and his numerical methods (and the progeny of ramanujan's numerical methods) are the closest we'll get. Which I think are second order factorial at best... He really changed the game for calculating \(\pi\). But also, I think this capped at double exponential--double factorial.
\[
f(x) = \sum_{n=0}^\infty \frac{x^n}{2^{2^{2^n}}}
\]
Has triple exponential convergence!
I'm not sure we're there yet. I think with ramanujan and his numerical methods (and the progeny of ramanujan's numerical methods) are the closest we'll get. Which I think are second order factorial at best... He really changed the game for calculating \(\pi\). But also, I think this capped at double exponential--double factorial.