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The best possible upper bound is the exact solution, and presumably the exact solution (and so the upper bound itself) will depend on the correlation $\rho$ between $X$ and $Y$.

We are given that $X \sim(N(0,1)$ and $Y \sim(N(0,1)$ where $X$ and $Y$ may be correlated, and let $\rho$ denote the correlation coefficient, $ -1 < \rho < 1$. This is all captured by a bivariate Normal pdf with joint density $f(x,y)$:

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

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:

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