  Monolayers (MLs) of transition metal dichalcogenides (TMDs) exhibit unusual
electrical behavior under magnetic fields due to their intrinsic spin-orbit
coupling and lack of inversion symmetry. While recent experiments have also
identified the critical role of carrier interactions within these materials, a
complete mapping of the ambipolar Landau level (LL) sequence has remained
elusive. Here, we use single-electron transistors to perform LL spectroscopy in
ML WSe$_2$, for the first time providing 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. This implies exceptionally strong many-body
interactions, and suggests that ML WSe$_2$ can serve as a host for new
correlated-electron phenomena.
