Mahler [Ma35] showed that this is $\gg \log\log n$. Schinzel [Sc67b] observed that for infinitely many $n$ it is $\leq n^{O(1/\log\log\log n)}$. The truth is probably $\gg (\log n)^2$ for all $n$.

The largest prime factors of $n(n+1)$ are listed as A074399 in the OEIS.