We find a remarkably enhanced production rate in star clusters (relative to the field) of very short-period, massive double white dwarf stars and of giant white dwarf binaries. These results are based on N-body simulations performed with the new GRAPE-6 special purpose hardware and are important in identifying and characterizing the progenitors of Type Ia supernovae. The high incidence of very close double white dwarf systems is the result of dynamical encounters between (mostly) primordial binaries and other cluster stars. Orbital hardening rapidly drives these degenerate binaries to periods under similar to10 hr. Gravitational radiation emission and mergers producing supra-Chandrasekhar objects follow in less than a Hubble time. If most stars are born in clusters, then estimates of the double white dwarf merger rates in galaxies (due to cluster dynamical interaction) must be increased more than 10-fold. A majority of the Roche lobe overflow giant white dwarf binaries are not primordial; they are produced in exchange reactions. Most cases result in a common envelope and formation of a double white dwarf binary rather than supersoft X-ray sources leading possibly to a Type Ia supernova.