Copper (Cu) as an excellent electrical conductor and the amphiphilic diblock copolymer polystyrene-block-poly(ethylene oxide) (PS-b-PEO) as a polymer electrolyte and ionic conductor can be combined with an active material in composite electrodes for polymer lithium-ion batteries (LIBs). As interfaces are a key issue in LIBs, sputter deposition of Cu contacts on PS-b-PEO thin films with high PEO fraction is investigated with in situ grazing-incidence small-angle X-ray scattering (GISAXS) to follow the formation of the Cu layer in real-time. We observe a hierarchical morphology of Cu clusters building larger Cu agglomerates. Two characteristic distances corresponding to the PS-b-PEO microphase separation and the Cu clusters are determined. A selective agglomeration of Cu clusters on the PS domains explains the origin of the persisting hierarchical morphology of the Cu layer even after a complete surface coverage is reached. The spheroidal shape of the Cu clusters growing within the first few nanometers of sputter deposition causes a highly porous Cu-polymer interface. Four growth stages are distinguished corresponding to different kinetics of the cluster growth of Cu on PS-b-PEO thin films: (I) nucleation, (II) diffusion-driven growth, (III) adsorption-driven growth, and (IV) grain growth of Cu clusters. Percolation is reached at an effective Cu layer thickness of 5.75 nm.