Let $F(n)$ be the prime in question. Pólya [Po18] proved that $F(n)\to \infty$ as $n\to\infty$. Mahler [Ma35] showed that $F(n)\gg \log\log n$. Schinzel [Sc67b] observed that for infinitely many $n$ we have $F(n)\leq n^{O(1/\log\log\log n)}$.

The truth is probably $F(n)\gg (\log n)^2$ for all $n$. Erdős [Er76d] conjectured that, for every $\epsilon>0$, there are infinitely many $n$ such that $F(n) <(\log n)^{2+\epsilon}$.

Pasten [Pa24b] has proved that \[F(n) \gg \frac{(\log\log n)^2}{\log\log\log n}.\] The largest prime factors of $n(n+1)$ are listed as A074399 in the OEIS.