For calculating
$$a \uparrow \uparrow n\ (mod\ m)$$
the chain
$$m , \phi(m) , \phi(\phi(m)) , \phi(\phi(\phi(m))) , ... $$
is useful.
As $\phi^n(m)=1$ for some n, the above modulo calculation gets stationary at some point.
So, it would be useful, if the length of the phi-chains could be suitable bounded.
Which lower and upper bounds are there for the least number n with $\phi^n(m)=1$ ?
I calculated the numbers up to about m = 36 * $10^6$ , and the longest chain consisted of 27 numbers until 1 was reached. The start number was the prime 35946497.
I'm not sure if there is a more recent reference than
P. Erdős, A. Granville, C. Pomerance and C. Spiro (1990). On the normal behavior of the iterates of some arithmetic functions. In Analytic number theory (pp. 165–204). Birkhäuser Boston.
http://www.math.dartmouth.edu/~carlp/iterate.pdf
The introduction cites a 1929 result of Pillai that
$$\left\lceil\frac{\log m}{\log 3}\right\rceil \le n \le \left\lceil\frac{\log m}{\log 2}\right\rceil,$$
so $n$ actually doesn't vary too wildly from $\log m$. The extreme values are occupied by powers of $3$ on the low end and powers of $2$ on the high end, so the paper is largely concerned with the normal order of $n$ (conjecturally, $n$ is asymptotic to a constant times $\log m$ for the vast majority of $m$).