(Joint work with David Dumas.) Convex real projective structures on surfaces, corresponding to discrete surface group representations into $\mathrm{PSL}(3, \mathbb{R})$, have associated to them affine spheres which project to the convex hull of their universal covers. Such an affine sphere is determined by its Pick (cubic) differential and an associated Blaschke metric (inducing a minimal map to $\mathrm{SL}(3,\mathbb{R})/SO(3))$. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial Pick cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons, and all polygons are obtained this way; along the way, a strong estimate on asymptotics is found.