>Given: $X \sim N(0,1)$ and $Y \sim N(0,1)$ where $X$ and $Y$ may be correlated.


If we assume that the underlying Normals are _jointly_ Normal, then the answer is quite simple. In particular, then $(X,Y)$ are bivariate Normal with correlation $\rho$ and joint pdf $f(x,y)$:


<img src="http://www.tri.org.au/se/bivariatejointpdfxyeqn.png">

Then, $E\big[Max[X,Y]\big]$ is:


<img src="http://www.tri.org.au/se/expectmaxxy.png">

where I am using the `Expect` function from the _mathStatica_ package for _Mathematica_ to automate the nitty gritties. Plainly, the expected maximum is contingent on $\rho$, as a quick plot illustrates:

<img src="http://www.tri.org.au/se/plotexpectmaxxy.png">

As @oferzeitouni noted, the maximum possible value is $\sqrt{\frac{2}{\pi}}$, which is attained when $\rho = -1$.