This amounts to finding the integral over the unit cube of $\max(x_1,x_2,...,x_n)$. Partition the unit cube into $n!$ equal parts according to how the magnitudes of the variables are ordered. Let us consider the part where $x_1>x_2>...>x_n$. For this part, we end up with the integral
$$\int_0^1 x_1\int_0^{x_1}\int_0^{x_2}...\int_0^{x_{n-1}}\,dx_n\,dx_{n-1}...dx_2\,dx_1.$$
This integral is equal to
$$\int_0^1 \frac{x_1^n}{(n-1)!}\,dx_1=\frac{1}{(n+1)(n-1)!}.$$
Multiply by $n!$ and you get $n/(n+1)$.