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 $f(x,y)$:
http://www.tri.org.au/se/bivariatejointpdfxyeqn.pngThen, $E\big[Max[X,Y]\big]$ is:
http://www.tri.org.au/se/expectmaxxy.pngwhere 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 $\rho = -1$.