The crucial early stages of amyloid growth, in which normally soluble proteins are converted into fibrillar nanostructures, are challenging to study using conventional techniques yet are critical to the protein aggregation phenomena implicated in many common pathologies. As with all nucleation and growth phenomena, it is difficult to track individual nuclei in traditional macroscopic experiments, which probe the overall temporal evolution of the sample, but do not yield detailed information on the primary nucleation step as they mix independent stochastic events into an ensemble measurement. To overcome this limitation, we have developed microdroplet assays enabling us to detect single primary nucleation events and to monitor their subsequent spatial as well as temporal evolution, both of which we find to be determined by secondary nucleation phenomena. By deforming the droplets to high aspect ratio, we visualize in real-time propagating waves of protein assembly emanating from discrete primary nucleation sites. We show that, in contrast to classical gelation phenomena, the primary nucleation step is characterized by a striking dependence on system size, and the filamentous protein self-assembly process involves a highly nonuniform spatial distribution of aggregates. These findings deviate markedly from the current picture of amyloid growth and uncover a general driving force, originating from confinement, which, together with biological quality control mechanisms, helps proteins remain soluble and therefore functional in nature.