A theorem of Zsigmondy says that for $n$ large enough (which means $n$ at least 7 in the worst case, for $r=2$), $r^n - 1$ has a prime factor $q$ which does not divide $r^m - 1$, for any $m$ less than $n$. So in fact something stronger holds. Let $f_n$ be the greatest prime factor of $r^n - 1$, then $f_n/n$ greater than $\log n$ eventually, as $f_n$ is greater than the $n$th prime for $n$ sufficiently large.

A theorem of Zsigmondy says that for $n$ large enough (which means $n$ at least 7 in the worst case, for $r=2$), $r^n - 1$ has a prime factor $q$ which does not divide $r^m - 1$, for any $m$ less than $n$. So in fact something stronger holds. Let $f_n$ be the greatest prime factor of $r^n - 1$, then $f_n/n$ greater than $\log n$ eventually, as $f_n$ is greater than the $n$th prime for $n$ sufficiently large.

(I see a missing piece, which I will try to repair. Of course, the largest prime factor among the factors of $r^m - 1$ for $m \lt n $ is greater than the $n$th prime, but I believe one can leverage this into a proof of the above.)