>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 (a) the exact answer is quite simple, and (b) we can do much better, conditioning the maximum bound on the correlation coefficient. In particular, if $(X,Y)$ are bivariate Normal with correlation $\rho$, then the joint pdf $f(x,y)$ is:


<img src="https://i.sstatic.net/gb2Ht.png">

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


<img src="https://i.sstatic.net/rxjme.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="https://i.sstatic.net/I2WzJ.png">

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