This link
https://primes.utm.edu/notes/gaps.html
gives a definition of the maximal gaps. For a number $g$ , $p(g)$ is the smallest prime $p$ followed by at least $g$ composites. The estimate is $\ln{p(g)}\approx \sqrt{g}$ and it is refined in the link.
But the approximation is far from tight comparing it with the values of the table.
- Is a better approximation known for $p(g)$ ?
- What are the tightest lower and upper bounds known for $p(g)$ ?