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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.