Is there a name for functions of this form $x_1^{x_2^{x_3\dots}}$?

Question

Or more specifically, is there a name for (and/or notation) used to characterize functions of the following form: $$\large x_1^{\Large x_2^{\large x_3^{\large x_4^{^\cdots}}}}$$ in which $x_1,x_2,x_3, x_4, \ldots$ etc are independent variables?

Answer 1

For the general case, such operation may be called simply nested exponentiation. Since your exponential is infinite, it would be infinite exponentiation. See terminology and this article for details.

As for the special case in which $x_k=x_m$ $\forall$ $k$,$m$, it is called tetration, the hyperoperation after exponentiation. It has apparently been much more developed than the general case.