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.

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.