Monolayers (MLs) of transition-metal dichalcogenides (TMDs) exhibit unusual electrical behaviour under magnetic fields due to their intrinsic spin-orbit coupling and lack of inversion symmetry(1-15). Although recent experiments have also identified the critical role of carrier interactions within these materials(11,15), a complete mapping of the ambipolar Landau level (LL) sequence has remained elusive. Here we use single-electron transistors (SETs)(16,17) to perform LL spectroscopy in ML WSe2, and provide a comprehensive picture of the electronic structure of a ML TMD for both electrons and holes. We find that the LLs differ notably between the two bands, and follow a unique sequence in the valence band (VB) that is dominated by strong Zeeman effects. The Zeeman splitting in the VB is several times higher than the cyclotron energy, far exceeding the predictions of a single-particle model and, moreover, tunes significantly with doping(15). This implies exceptionally strong many-body interactions, and suggests that ML WSe2 can serve as a host for new correlated-electron phenomena.