Forests emit large quantities of volatile organic compounds (VOCs) to the atmosphere. Their condensable oxidation products can form secondary organic aerosol, a significant and ubiquitous component of atmospheric aerosol, which is known to affect the Earth's radiation balance by scattering solar radiation and by acting as cloud condensation nuclei. The quantitative assessment of such climate effects remains hampered by a number of factors, including an incomplete understanding of how biogenic VOCs contribute to the formation of atmospheric secondary organic aerosol. The growth of newly formed particles from sizes of less than three nanometres up to the sizes of cloud condensation nuclei (about one hundred nanometres) in many continental ecosystems requires abundant, essentially non-volatile organic vapours, but the sources and compositions of such vapours remain unknown. Here we investigate the oxidation of VOCs, in particular the terpene α-pinene, under atmospherically relevant conditions in chamber experiments. We find that a direct pathway leads from several biogenic VOCs, such as monoterpenes, to the formation of large amounts of extremely low-volatility vapours. These vapours form at significant mass yield in the gas phase and condense irreversibly onto aerosol surfaces to produce secondary organic aerosol, helping to explain the discrepancy between the observed atmospheric burden of secondary organic aerosol and that reported by many model studies. We further demonstrate how these low-volatility vapours can enhance, or even dominate, the formation and growth of aerosol particles over forested regions, providing a missing link between biogenic VOCs and their conversion to aerosol particles. Our findings could help to improve assessments of biosphere-aerosol-climate feedback mechanisms, and the air quality and climate effects of biogenic emissions generally.