I am working with some tetration problems, such as below: $$y = e^{e^x}$$
and I am looking for a concise notation for this. In particular, I would like a way to indicate $n$ iterations of the exponentiation, with the deepest level raised to $x$, rather than $e$.
My initial thoughts were to write the above example as $e^{e^x} = ({^2e})^x = (e\uparrow\uparrow2)^x$. However, since exponential towers must be evaluated from top to bottom, it seems like this is not true.
Is there any other concise notation for this?
Thanks.
(Replacing my earlier comments to make a proper answer)
Suggestion 1: In the tetration forum we have partly used $\exp^{\circ h}_b(x)$ (the little circle indicating function-composition instead of powers or instead of derivation) for the general iteration $x \to b^x$ to the iteration-(h)eight $h$ of the exponentialtower. I myself use sometimes $\text{T}^{\circ h}_b(x)$ for shortness and $\text{U}^{\circ h}_b(x)$ for the decremented exponentiation $x \to b^x−1 $.
Suggestion 2: In many articles I've also seen the simple solution to use the index-notation. So $z_0$ for the initial value , $z_1=b^{z_0}$ then $z_h=b^{z_{h−1}}$ for the $h$'th iteration (exponentialtower of (h)eight $h$) and $z_\infty$ if that limit exists. In articles the base $b$ is mostly a fixed parameter over a lot of formulae and algebraic derivations so I'd prefer such a notation which allows to omit this reference to $b$ to reduce redundancy in notation. (I find this unbeatable concise - unfortunately the indexing-notation indicates many things in math so I use this only when I'm well sure it is not obfuscating my line of discussion/derivation/definition)