We study the cosmology of a toy modified theory of gravity in which gravity shuts off at short distances, as in the fat graviton scenario of Sundrum. In the weak-field limit, the theory is perturbatively local, ghost-free and unitary, although likely suffers from non-perturbative instability. We derive novel self-inflationary solutions from the vacuum equations of the theory, without introducing scalar fields or other form of stress energy. The modified perturbation equation expressed in terms of the Newtonian potential closely resembles its counterpart for inflaton fluctuations. The resulting scalar spectrum is therefore slightly red, akin to the simplest scalar-driven inflationary models. A distinguishing feature from standard inflation, however, is a blue spectrum of gravitational waves. The specific model considered here yields such a strong blue tilt that the tensor amplitude on large scales is orders of magnitude below detectable levels, analogous to the predictions of ekpyrotic/cyclic scenarios.