We present the results from a three-night, time-series study of the open cluster NGC 6791 using the Megacam wide-field mosaic CCD camera on the 6.5 m MMT telescope. The aim of this study was to demonstrate the ability to obtain very high precision photometry for a large number of stars. We achieved better than 1% precision for more than 8000 stars with 14.3<R<20.1 and submillimagnitude (as low as 0.36 mmag) precision for over 300 stars with 14.6<R<16.3 in the field of this cluster. We also discovered 10 new variable stars, including a possible δ Scuti variable with an amplitude of 2%, eight likely W UMa contact binaries, and a possible RS CVn star, and we identified seven suspected low-amplitude variables, including one star with an amplitude of 4 mmag. We comment on the implications of this study for a ground-based survey for transiting planets as small as Neptune.