Summation series uses the Sigma symbol like this.
$$\sum_{n=0}^{\infty} \frac{1}{2^n} = 2$$
Product Series uses the Pi symbol like this.
$$\prod_{n=1}^{\infty} \frac{4n^2}{4n^2 - 1} = \frac{\pi}{2}$$
What about exponentiation series? Does this exist? What symbol is used for it? E.g., the series starts at n=1 and goes to infinity, and the term prototype is just n, would yield this:
$$1^{2^{3^{...}}}$$
Obviously that doesn't converge, but I just wanted to show some example.
Are exponentiation series studied in math? I would like to read more about them.
Note: I saw the tag for power series, but it does not appear to be what I want. E.g., this question is a summation series.
While there is no ''standard'' notation, as you describe, that is analogous to the sigma and pi notations, you may use Knuth's up-arrow notation instead.
This notation uses $\uparrow$ to denote exponentiation, $\uparrow\uparrow$ do denote tetration, $\uparrow\uparrow\uparrow$ to denote pentation, and so forth. For instance, $$ {\displaystyle 2\uparrow 4=2\times (2\times (2\times 2))=2^{4}=16}, $$ and $$ {\displaystyle 2\uparrow \uparrow 4=2\uparrow (2\uparrow (2\uparrow 2))=2^{2^{2^{2}}}=65536}. $$