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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?

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    This link https://en.wikipedia.org/wiki/Tetration may be useful. – 2017-02-14
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    Sequences in powers are highly non-trivial. – 2017-02-14
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    David Francis Barrow (1888-1970) published a study of these functions titled [*Infinite exponentials*](http://www.math.usm.edu/lee/BarrowInfiniteExponentials.pdf) [American Mathematical Monthly 43 #3 (March 1936), pp. 150-160], where he proved some basic convergence results in analogy with the kinds of convergence results one can prove for infinite series, infinite products, and continued fractions. Others have have also worked on this topic, both before and after Barrow. – 2017-02-15

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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.

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    Actually, Knuth's up-arrow is only for $x_n=x_k$ for all $n,k$. – 2017-02-14
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    Yes, it is only for the first form asked about on the question. – 2017-02-14
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    As your first link clearly states, tetration of $a\;n $ times is denoted by $\Large ^na = a^{a^{a^{a ...}}}$ – 2017-02-15
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    The OP's question is asking about the name for $$\Large a^{b^{c^{d \cdots}}}$$ where it may be true that all or some of $a, b, c, d, \cdots$ are distinct. – 2017-02-15
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    So unless you want to extend/qualify the meaning of tetration, this answer is wrong. – 2017-02-15
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    The OP's question asks about the the name of the function $f(x_1, x_2, x_3, \ldots, x_n) = \Large {x_1}^{{x_2}^{{x_3}^{{\cdots}^{x_n}}}}$. – 2017-02-15
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    You are right, of course - OP edited the question, which was originally about the function $f(x) =x^{x^{x^{x^{...}}}}$. Therefore, my answer will be edited as well. – 2017-02-15