The importance of understanding the mechanism of protein aggregation into insoluble amyloid fibrils lies not only in its medical consequences, but also in its more basic properties of self-organization. The discovery that a large number of uncorrelated proteins can form, under proper conditions, structurally similar fibrils has suggested that the underlying mechanism is a general feature of polypeptide chains. In this work, we address the early events preceding amyloid fibril formation in solutions of zinc-free human insulin incubated at low pH and high temperature. Here, we show by time-lapse atomic force microscopy that a steady-state distribution of protein oligomers with a quasiexponential tail is reached within a few minutes after heating. This metastable phase lasts for a few hours, until fibrillar aggregates are observable. Although for such complex systems different aggregation mechanisms can occur simultaneously, our results indicate that the prefibrillar phase is mainly controlled by a simple coagulation-evaporation kinetic mechanism, in which concentration acts as a critical parameter. These experimental facts, along with the kinetic model used, suggest a critical role for thermal concentration fluctuations in the process of fibril nucleation.