The average gap is log(n), but the maximal gap is not well understood.
According to https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture, current results are
Unconditional: n^0.525
Conditional on RH: log(n)*sqrt(n)
Conjecturally: log(n)^2
According to https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture, current results are
Unconditional: n^0.525
Conditional on RH: log(n)*sqrt(n)
Conjecturally: log(n)^2
That's quite fascinating.