The universe may have extra spatial dimensions with large volume that we cannot perceive because the energy required to excite modes in the extra directions is too high. Many examples are known of such manifolds with a large volume and a large mass gap. These compactifications can help explain the weakness of four-dimensional gravity and, as we show here, they also have the capacity to produce reasonable potentials for an inflaton field. Modeling the inflaton as a bulk scalar field, it becomes very weakly coupled in four dimensions, and this enables us to build phenomenologically acceptable inflationary models with tunings at the few per mil level. We speculate on dark matter candidates and the possibility of braneless models in this setting.