Is there some accepted, more concise notation for expressions like $\log \log \log n$?
I just noticed [an arXiv posting](http://arxiv.org/abs/1412.5029) 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.



Notation for $\log \log \cdots \log n$?
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](http://www.math.uiuc.edu/~ford/wwwpapers/primegaps.pdf) by Ford, Green, Konyagin, and Tao. Alternatively, Claudia Spiro used $L\_4x$ for $\log\log\log\log x$ in [this paper](http://matwbn.icm.edu.pl/ksiazki/aa/aa68/aa6842.pdf). 


