According to Wikipedia,
For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2.
Is it really necessary to specify here, that the faces are also simply connected?
I can imagine a simply connected body with a not simply connected face (for example a cylinder), but can't imagine a simply connected polyhedron with a not simply connected face. Does such thing exist at all?
Consider the following construction. Take a solid tetrahedron, and attach a very small tetrahedron to the middle of one of the faces. The boundary of this object is simply connected (it's a sphere, topologically), but one of the faces looks like a triangle with a small triangle removed from its center, which is not simply connected (it's an annulus, topologically).