Motivated by recent developments in wireless power transfer, we study communication with a remotely powered transmitter. We propose an information-theoretic model where a charger can dynamically decide on how much power to transfer to the transmitter based on its side information regarding the communication, while the transmitter needs to dynamically adapt its coding strategy to its instantaneous energy state, which in turn depends on the actions previously taken by the charger. We characterize the capacity as an $n$-letter mutual information rate under various levels of side information available at the charger. When the charger is finely tunable to different energy levels, referred to as a "precision charger", we show that these expressions reduce to single-letter form and there is a simple and intuitive joint charging and coding scheme achieving capacity. The precision charger scenario is motivated by the observation that in practice the transferred energy can be controlled by simply changing the amplitude of the beamformed signal. When the charger does not have sufficient precision, for example when it is restricted to use a few discrete energy levels, we show that the computation of the $n$-letter capacity can be cast as a Markov decision process if the channel is noiseless. This allows us to numerically compute the capacity for specific cases and obtain insights on the corresponding optimal policy, or even to obtain closed form analytical solutions by solving the corresponding Bellman equations, as we demonstrate through examples. Our findings provide some surprising insights on how side information at the charger can be used to increase the overall capacity of the system.