We present the first results from our Hubble Space Telescope brightest cluster galaxy (BCG) survey of seven central supergiant cluster galaxies and their globular cluster (GC) systems. We measure a total of 48,000 GCs in all seven galaxies, representing the largest single GC database. We find that a log-normal shape accurately matches the observed the luminosity function (LF) of the GCs down to the globular cluster luminosity function turnover point, which is near our photometric limit. In addition, the LF has a virtually identical shape in all seven galaxies. Our data underscore the similarity in the formation mechanism of massive star clusters in diverse galactic environments. At the highest luminosities (L ≳ 107L⊙), we find small numbers of "superluminous" objects in five of the galaxies; their luminosity and color ranges are at least partly consistent with those of ultra-compact dwarfs. Last, we find preliminary evidence that in the outer halo (R ≳ 20 kpc), the LF turnover point shows a weak dependence on projected distance, scaling as L0 ∼ R-0.2, while the LF dispersion remains nearly constant.