Using an efficient cluster approach, we study the physics of two-dimensional lattice bosons in a strong magnetic field in the regime where the tunneling is much weaker than the on-site interaction strength. We study both the dilute, hard-core bosons at filling factors much smaller than unity occupation per site and the physics in the vicinity of the superfluid-Mott lobes as the density is tuned away from unity. For hard-core bosons, we carry out extensive numerics for a fixed flux per plaquette φ=1/5 and φ=1/3. At large flux, the lowest-energy state is a strongly correlated superfluid, analogous to He-4, in which the order parameter is dramatically suppressed, but nonzero. At filling factors ν=1/2,1, we find competing incompressible states which are metastable. These appear to be commensurate density wave states. For small flux, the situation is reversed and the ground state at ν=1/2 is an incompressible density wave solid. Here, we find a metastable lattice supersolid phase, where superfluidity and density wave order coexist. We then perform careful numerical studies of the physics near the vicinity of the Mott lobes for φ=1/2 and φ=1/4. At φ=1/2, the superfluid ground state has commensurate density wave order. At φ=1/4, incompressible phases appear outside the Mott lobes at densities n=1.125 and n=1.25, corresponding to filling fractions ν=1/2 and 1, respectively. These phases, which are absent in single-site mean-field theory, are metastable and have slightly higher energy than the superfluid, but the energy difference between them shrinks rapidly with increasing cluster size, suggestive of an incompressible ground state. We thus explore the interplay between Mott physics, magnetic Landau levels, and superfluidity, finding a rich phase diagram of competing compressible and incompressible states. © 2016 American Physical Society.