Amorphous silicon is a promising high-capacity anode material for application in lithium-ion batteries. However, a huge drawback of the material is that the large capacity losses taking place during cycling lead to an unstable performance. In this study we investigate the capacity losses occurring during galvanostatic lithiation of amorphous silicon thin film electrodes by in-situ neutron reflectometry experiments for the first ten cycles. As determined from the analysis of the neutron scattering length density and of the film thickness, the capacity losses are due to irreversible storage of lithium in the electrode. The amount of stored lithium increases during cycling to 20% of the maximum theoretical capacity after the 10th cycle. Possible explanations are discussed.