  We report the first electronic compressibility measurements of magic-angle
twisted bilayer graphene. The evolution of the compressibility with carrier
density offers insights into the interaction-driven ground state that have not
been accessible in prior transport and tunneling studies. From capacitance
measurements, we determine chemical potential as a function of carrier density
and find the widths of the energy gaps at fractional filling of the moir\'{e}
lattice. In the electron-doped regime, we observe unexpectedly large gaps at
quarter- and half-filling and strong electron-hole asymmetry. Moreover, we
measure a $\mathord{\sim}35\,\textrm{meV}$ mini-bandwidth that is much wider
than most theoretical estimates. Finally, we explore the field dependence up to
the quantum Hall regime and observe significant differences from transport
measurements.
