At this MathOverflow post there is an interesting comment below the O.P.'s question. The comment is due to Greg Martin and it basically reads that
Allmost all integers have the property that $\text{gcd}(n,\phi (n))$ is equal to the product of all prime powers dividing $n$, which are also less than $\log \log n.$
Question: I was thinking about it, but how can we get this $\log\log n$ bound? Do you have any hints or references for the proof? I believe that very elementary techniques don't work.
Motivation: If I will understand how this result is derived, I will also realize how we can intuitively guess that (using arguments of probabilistic nature)
$$\#\{n\leq x,\ \text{gcd}(n,\phi (n))=1 \}\sim x\prod_{p\leq \log \log x}\left(1-\frac{1}{p}\right),\ x\rightarrow +\infty,\ \ \ (1)$$
where $n$ stands for a positive integer and $p$ denotes a prime number. Using Merten's theorem, $(1)$ then becomes what Erdos proved here.