The onset of lasing in a `classical' laser is marked by a threshold that can be characterized by a sharp increase in photon flux as a function of external pumping strength. The same is not necessarily true for a single trapped atom interacting with a single optical mode. `Quantum' lasers have shown thresholdless lasing in the regime of strong coupling between an atom and a radiation field, but the existence of a threshold has been predicted. Here, we demonstrate and characterize a single-atom laser with and without threshold behaviour by changing the strength of atom/light-field coupling. We observe the establishment of a laser threshold through the accumulation of photons in the optical mode even for a mean photon number substantially lower than for the classical case. Our observations mark an important step towards a fundamental understanding of laser operation in the few-atom limit including systems based on semiconductor quantum dots or molecules.