We consider a two-dimensional system of electrons interacting via an infinitely strong $\delta$-function repulsion and argue that this is an archetype for non-Fermi liquid physics. We represent the electrons as composite bosons interacting with a Chern-Simons gauge field, which enables us to construct a dual description in which the fundamental constituents are vortices in the auxiliary boson fields. The resulting formalism embraces a cornucopia of possible phases. Remarkably, superconductivity is a generic feature, while the Fermi liquid is not -- prompting us to conjecture that such a state may not be possible for this model. Many aspects of our earlier discussions of the nodal liquid and spin-charge separation find surprising incarnations in this new framework.