We present BVRI photometry and optical spectroscopy of two Type Ic supernovae SN 2007bg and SN 2007bi discovered in wide-field, non-targeted surveys and associated with sub-luminous blue dwarf galaxies. Neither SNe 2007bg nor 2007bi were found in association with an observed GRB, but are found to inhabit similar low-metallicity environments as GRB associated supernovae. The radio-bright SN 2007bg is hosted by an extremely sub-luminous galaxy of magnitude MB = -12.4+/-0.6 mag with an estimated oxygen abundance of 12+log(O/H) = 8.18+/-0.17. The lightcurve of SN 2007bg displays one of the fastest post-maximum decline rates of all broad-lined Type Ic supernovae known to date and, when combined with its high expansion velocities, a high kinetic energy to ejected mass ratio (E_K/Mej ~ 2.7). We show that SN 2007bi is possibly the most luminous Type Ic known, reaching a peak magnitude of MR ~ 21.3 mag and displays a remarkably slow decline, following the radioactive decay rate of 56Co to 56Fe throughout the course of its observed lifetime. From a simple model of the bolometric light curve of SN 2007bi we estimate a total ejected 56Ni mass of M_Ni = 3.5 - 4.5 solar masses, the largest 56Ni mass measured in the ejecta of a supernova to date. There are two models that could explain the high luminosity and large ejected 56Ni mass. One is a pair-instability supernova (PISN) which has been predicted to occur for massive stars at low metallicities. We measure the host galaxy metallicity of SN 2007bi to be 12 + log(O/H) = 8.15+/-0.15 which is somewhat high to be consistent with the PISN model. An alternative is the core-collapse of a C+O star of 20 - 40 solar masses which is the core of a star of originally 50 - 100 solar masses. (Abridged)
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