Energy Harvesting allows the devices in a Wireless Sensor Network to recharge their batteries through environmental energy sources. While in the literature the main focus is on devices with ideal batteries, in reality several inefficiencies have to be considered to correctly design the operating regimes of an Energy Harvesting Device (EHD). In this work we describe how the throughput optimization problem changes under \emph{real battery} constraints in an EHD. In particular, we consider imperfect knowledge of the state of charge of the battery and storage inefficiencies, \emph{i.e.}, part of the harvested energy is wasted in the battery recharging process. We formulate the problem as a Markov Decision Process, basing our model on some realistic observations about transmission, consumption and harvesting power. We find the performance upper bound with a real battery and numerically discuss the novelty introduced by the real battery effects. We show that using the \emph{old} policies obtained without considering the real battery effects is strongly sub-optimal and may even result in zero throughput.