When is iterated exponentiation used and how is it defined?

I was thinking of ways to define an iterated exponentiation operation. The nice thing about addition and multiplication is that they're associative and commutative, which makes defining the sum and product operations quite unambiguous. Exponentiation is neither associative nor commutative, so there are many ways you could define an iterated version. I have defined down below what I think are the 4 most obvious candidates.

$$\def\ite#1#2#3#4{\vcenter{\overset{#3}{\underset{#2}{\LARGE #1}}}~{#4}} \begin{array}{ccccc} \ite{\mathrm E}{n=b}{a}{x_n}&=&x_b^{\left(\ite{\mathrm E}{n=b+1}{a}{x_n}\right)}&&\\ \ite{E}{n=b}{a}{x_n}&=&x_a^{\left(\ \ite{E}{n=b}{a-1}{x_n}\right)}&&\\ \ite{\mathcal E}{n=b}{a}{x_n}&=&\left(\ite{\mathcal E}{n=b+1}{a}{x_n}\right)^{x_b}&=&x_a^{\prod_{n=b}^{a-1} x_n}\\ \ite{\mathscr E}{n=b}{a}{x_n}&=&\left(\ \ite{\mathscr E}{n=b}{a-1}{x_n}\right)^{x_a}&=&x_b^{\prod_{n=b+1}^{a} x_n}\\ \end{array}$$

As has been displayed, the notations $\mathcal E$ and $\mathscr E$ are quite redundant as they can both be expressed in terms of a product.

From the few test cases I've checked, $E$ does not take kindly to limits and appears to favour converging to 1 or diverging. Does $E$ see any use in mathematics, and if so, by what name?

Finally, $\mathrm E$ plays rather nice. For example:

$$\ite{\mathrm E}{n=1}{\infty}{\frac{n+1}{n}}=\frac{2}{1}^{\frac{3}{2}^{\frac{4}{3}^{\frac{5}{4}^⋰}}}\approx 3.5038099724520166$$

Does $\mathrm E$ have a common name or symbol? In which areas of mathematics can it be found? If neither $E$ nor $\mathrm E$ is commonly used, what definition is? Or, is this sort of operation just uninteresting or useless?

• As far as I know tetration is defined as itered exponentation – Alessandro Codenotti Nov 18 '14 at 13:55
• @Alessandro Yes, this is true. But multiplication is to summation as tetration is to what I am looking for, where each term is part of a sequence and not always the same number. – Regret Nov 18 '14 at 13:58