The expected value is asymptotic to $(\log n)/n$ as $n$ tends to infinity. One way to see this is to use the representation of order statistics of uniform points as the first $n$ points of a Poisson process, normalized by the $n+1$ Point. Since the sum of $n+1$ exponential variables is concentrated, the question reduces to the distribution of the maximum of $n-1$ Exponential variables.

The expected value is asymptotic to $(\log n)/n$ as $n$ tends to infinity (By "asymptotic" I mean that the ratio tends to 1). One way to see this is to use the representation of order statistics of uniform points as the first $n$ points of a Poisson process, normalized by the $n+1$ Point. Since the sum of $n+1$ exponential variables is concentrated, the question reduces to the distribution of the maximum of $n-1$ Exponential variables.

Usually one considers the maximum $M_n$ of the $n+1$ gaps including the gap between zero and the first point and between 1 and the last point, but that does not change the asymptotics. The known properties of $M_n$ are surveyed in the introduction of Devroye's paper that Brendan McKay mentioned: https://projecteuclid.org/download/pdf_1/euclid.aop/1176994313

In particular the fact about Poisson processes I noted above is Lemma 2.1 there, and lemmas 2.4, 2.5 and 2.6 describe more precise asymptotics for the maximal gap.