For independently distributed $x_i$'s, each with cumulative distribution $$F_i(x_i)=\tfrac{1}{2}+\tfrac{1}{2}{\rm Erf}\,[(x_i-\mu_i)/(\sigma_i\sqrt 2],$$ 
the cumulative distribution of the maximum is given by
$$P({\rm max}_i \,x_i<X_{\rm max})=\prod_{i=1}^n P(x_i<X_{\rm max})=\prod_{i=1}^n F_i(X_{\rm max}).$$
For small $n$ you can now calculate moments of $X_{\rm max}$ by integration,
$$E(X_{\rm max}^p)=\int_{-\infty}^\infty x^p\frac{d}{dx}\left(\prod_{i=1}^n F_i(x)\right)\,dx.$$
There is unlikely to be a closed-form answer for arbitrary $n$, in fact, even the $n=2$ integral seems problematic (*Mathematica* fails to evaluate it). If you take the $\mu_i$'s and $\sigma_i$'s to be the same, then progress can be made, for $n=2$ I find
$$E(X_{\rm max})=\mu+\sigma/\sqrt\pi,\;\;{\rm Var}\,(X_{\rm max})=(1-1/\pi)\sigma^2.$$

Perhaps you are satisfied with a *large-$n$ approximation*. For identical $\mu_i$'s and $\sigma_i$'s it is given by the <A HREF="https://en.wikipedia.org/wiki/Fisher–Tippett–Gnedenko_theorem">Fisher–Tippett–Gnedenko theorem</A>, see for example  <A HREF="https://math.stackexchange.com/questions/89030/expectation-of-the-maximum-of-gaussian-random-variables">this MSE posting.</A> I have not yet found the generalization to arbitrary $\mu_i$'s and $\sigma_i$'s.