The fatness of a 4-polytope or 3-sphere is defined as $(f_1+f_2-20)/(f_0+f_3-10)$. We construct arbitrarily fat, strongly regular CW 3-spheres that are both shellable and dual shellable. These spheres have $f$-vectors $(Θ(n),Θ(nα(n)),Θ(nα(n)),Θ(n))$, where $α$ is the inverse Ackermann function.