For a closed affine manifold M of dimension m the developing map defines an open subset D(M) C Rm. We show that D(M) cannot lie between parallel hyperplanes. When m .;;; 3 we show that any nonconstant polynomial p: R m --> R is unbounded on D( M). If D( M) lies in a half-space we show M has zero Euler characteristic. Under various special conditions on M we show that M has no nonconstant functions given by polynomials in affine coordinates.