Strongly driving a two-level quantum system with light leads to a ladder of Floquet states separated by the photon energy. Nanoscale quantum devices allow the interplay of confined electrons, phonons, and photons to be studied under strong driving conditions. Here, we show that a single electron in a periodically driven double quantum dot functions as a "Floquet gain medium," where population imbalances in the double quantum dot Floquet quasienergy levels lead to an intricate pattern of gain and loss features in the cavity response. We further measure a large intracavity photon number n(c) in the absence of a cavity drive field, due to equilibration in the Floquet picture. Our device operates in the absence of a dc current-one and the same electron is repeatedly driven to the excited state to generate population inversion. These results pave the way to future studies of nonclassical light and thermalization of driven quantum systems.