Maximum entropy moment closure systems of gas dynamics are investigated. It is shown that polynomial weight functions growing superquadratically at infinity lead to hyperbolic systems with an unpleasant state space: equilibrium states are boundary points with possibly singular fluxes. This in its generality previously unknown result applies to any moment system including, for example, the 26 or 35 moment case. One might try to avoid singular fluxes by choosing non-polynomial weight functions which grow sub-quadratically at infinity. This attempt, however, is shown to be incompatible with the Galilean invariance of the moment systems because rotational and translational invariant, finite dimensional function spaces necessarily consist of polynomials.