I wonder if a smooth manifolds $M$ which admits an affine structure must necessarily satisfy $\pi\_2(M)=0$.


By affine structure I mean an atlas all of whose change of coordinates maps are affine maps. This kind of manifolds occur as bases of Lagrangian fibrations. If the answer to the above question is yes, this would imply that the fibers of a Lagrangian fibration are always incompressible (their $\pi\_1$ injects in the total space).



Smooth manifold with affine structure: aspherical?
No they are not, you can consider $\mathbb{R}^3-\{0\}$ or the quotient of $\mathbb{R}^3-\{0\}$ by the map $h\_c(x)=cx, c>1$ which is $S^2\times S^1$. 


