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.