Disclaimer: I know very little about Shimura varieties.
Some Shimura varieties have a contractible universal covering space, for instance $A_g$ itself. Are there any nice necessary and/or sufficient conditions implying this?
Connected Shimura varieties are quotients $S=X/\Gamma$, where $X$ is a Hermitian symmetric space without compact factor and $\Gamma$ is a discrete subgroup acting properly discontinuously on $X$. If $\Gamma$ acts without fixed points, then $X\to S$ is a universal covering. (In the general case, the universal covering of $S$ is $X/\Gamma_1$, where $\Gamma_1$ is the subgroup of $\Gamma$ generated by stabilizers of fixed points on $X$.)
It is known that $X=G/K$, where $G$ is a semisimple adjoint Lie group and $K$ is a maximal compact subgroup. The Iwasawa decomposition $G=KAN$ implies that $X$ is diffeomorphic to $AN$, hence to a real vector space. In particular, $X$ is contractible.