How to deal with Multiple Logs? (Big O)
The starting point is the limit
$$
\lim_{n\to\infty} \frac{\log n}{n} = 0,
$$
which can also be stated succinctly as $\log n = o(n)$. If $f(n)$ is any function such that $\lim_{n\to\infty} f(n) = \infty$, it then follows that
$$
\lim_{n\to\infty} \frac{\log f(n)}{f(n)} = 0,
$$
or more succinctly, $\log f(n) = o(f(n))$. In particular, taking $f(n) = \log n$ we get $\log\log n = o(\log n)$, taking $f(n) = \log\log n$ we get $\log\log\log n = o(\log\log n)$, and so on.

More generally, if $a(n) = o(b(n))$ and $f(n) \to \infty$ then $a(f(n)) = o(b(f(n))$. (The case above was $a(n) = \log n$ and $b(n) = n$.) There are similar rules for big O, which I leave for the reader to discover.