We study the problem of hedging early exercise (American) options with respect to exponential utility within a general incomplete market model. This leads us to construct a duality formula involving relative entropy minimization and optimal stopping. We further consider claims with multiple exercises, and static-dynamic hedges of American claims with other European and American options. The problem is important for accurate valuation of employee stock options (ESOs), and we demonstrate this in a standard diffusion model. We find that incorporating static hedges with market-traded options induces the holder to delay exercises and increases the ESO cost to the firm.