  We extend the standard Bose-Hubbard model to capture arbitrarily strong
on-site correlations. In addition to being important for quantitatively
modeling experiments, for example, with Rubidium atoms, these correlations must
be included to describe more exotic situations. Two such examples are when the
interactions are made large via a Feshbach resonance, or when each site rotates
rapidly, making a coupled array of quantum Hall puddles. Remarkably, even the
mean field approximation to our model includes all on-site correlations. We
describe how these on-site correlations manifest themselves in the system's
global properties: modifying the phase diagram and depleting the condensate.
