This is a result of Pillai. Indeed we have $\text{rphi}(n)=\frac{\log n}{\log 2}$ when $n$ is a power of 2, and we have $\text{rphi}(n)=\lceil\frac{\log n}{\log 3}\rceil$ when $n$ is twice a power of 3. Pillai proved that these two cases characterise the extremal behaviour of $\text{rphi(n})$.

$$\lceil\frac{\log n}{\log 2}\rceil\geq\text{rphi}(n)\geq\lceil\frac{\log n}{\log 3}\rceil$$
If one considers the numbers $n=2^a3^b$ one can see that $\frac{\text{rphi}(n)}{\log n}$ is dense in $[\frac{1}{\log 3},\frac{1}{\log 2}]$.

"On a function connected with ϕ(n)" S. S. Pillai, Bull. Amer. Math. Soc. Volume 35, Number 6 (1929), 837-841.