Continuum sum - a new hope

[bo198214]

Why does what Ansus wrote fails for a = e, for instance?

For regular iteration you need a fixed point.
Real fixed points exist only for base \( b<e^{1/e} \).
Of course you can do it also at a complex fixed point of \( e^x \) however then the resulting tetration is not real on the real axis (actually it is not even analytic).