Notation for $\log \log \cdots \log n$? Is there some accepted, more concise notation for expressions like $\log \log \log n$?
I just noticed an arXiv posting that quotes the bound
$$
\frac{\log X \log \log X \log \log \log \log X}
{ \log \log \log X}
$$
and I wonder if there is room for some notational improvement in this domain.
Clearly $\log^k n$ is not an option, as that seems entrenched in the literature
to mean $(\log n)^k$.
If there is no accepted, more concise notation, perhaps the community could invent
a notation.
 A: Analytic Number Theorists have been using $\log_nx$ for the $n$-times iterated (natural) logarithm of $x$ for some time. See, e.g., the first page of this paper by Ford, Green, Konyagin, and Tao. Alternatively, Claudia Spiro used $L_4x$ for $\log\log\log\log x$ in this paper. 
A: I find $(\text{log}^k)(n)$ quite clear, with little or no potential for misunderstanding.
A: I have seen the notation $F^{\circ2}(x)$ and $F^{\circ n}(x)$ in a Dynamical Systems book by Shlomo Sternberg (see e.g. section 2.2.2 Period doubling, there). Clearly $\circ$ indicates that we are talking about composition of functions and distinguishes it from $F^2(x)$ and $F^n(x)$ if you interpret the latter as $(F(x))^2$ and $(F(x))^n$. So one might perhaps use $\log^{\circ4}n$ and $\log^{\circ k}n$. I would personally prefer this, since $\log_4n$ could be confused with $\log$ with base $4$, though I am sure people in analytic number theory know what they mean with their choice of notation.  
A: I don't know about generally accepted, but $\log^{(n)}X$ seems pretty concise. It conforms to the usual notation for functional iterations and there is no room for confusion with either taking powers or using different bases, e.g. $(\log_2^{(n)}X)^k$.
