Transseries, nest-series, and other exotic series representations for tetration

[bo198214]

So far, I've found 2 types of transseries representation such that if the series converges, the continuum sum does as well (proof given earlier here):

I was a bit puzzled why \( e^{e^x} \) converges via transseries while \( e^{x^2} \) does not converge. Actually \( e^{\frac{x^2}{2}} \) must also converge as we have \( \frac{x^2}{2}<e^x \), and indeed it also converges! But it seems that this already the limit, i.e. every series \( e^{c x^2} \) for \( c>\frac{1}{2} \) does not converge. I guess that this is the same for \( e^{\frac{x^n}{n!} \), \( n>1 \), while Faulhaber always works for \( n=1 \), i.e. \( e^{c x} \). Do you have any idea how to sum \( e^{x^2} \)?