Superroots and a generalization for the Lambert-W

[Gottfried]

The thing is solving (x_m ^ x_m)^[m] = y is only close to solving
X_n^^[n] = y ( n = m in value )

True. But having this way a (non-trivial) vector of different exponents (or better: bases) which comes out to be a meaningful "nested exponentiation" I'm curious, whether one can do something with it, for instance weighting, averaging, or multisecting that sequence of exponents/bases when re-combining them to a "nested exponential". We have not yet many examples of "nested exponentiations" with a meaningful outcome.
For instance, the construction of the Schroeder-function is based on (ideally) infinite iteration of the base-function to get a linearization. If we iterate the h2()-function infinitely, the curve of the consecutive values in an x/y-diagram (where x is the iteration number) approach a horizontal line; don't know whether using that linearization shall prove useful for something similar.

(When Euler found his version of the gamma-function, that was in one version putting together sequences of integer numbers weighting and repeating in a meaningful way; there is some infinite product-representation for his gamma-function I think I recall correctly... )

(see also the updates in my previous (introducing) posting)