We present new deep photometry of the globular cluster system (GCS) around NGC 6166, the central supergiant galaxy in Abell 2199. Hubble Space Telescope data from the Advanced Camera for Surveys and WFC3 cameras in F475W and F814W are used to determine the spatial distribution of the GCS, its metallicity distribution function (MDF), and the dependence of the MDF on galactocentric radius and on GC luminosity. The MDF is extremely broad, with the classic red and blue subpopulations heavily overlapped, but a double-Gaussian model can still formally match the MDF closely. The spatial distribution follows a Sérsic-like profile detectably to a projected radius of at least Rgc = 250 kpc. To that radius, the total number of clusters in the system is NGC = 39000 ± 2000, the global specific frequency is SN = 11.2 ± 0.6, and 57% of the total are blue, metal-poor clusters. The GCS may fade smoothly into the intracluster medium (ICM) of A2199; we see no clear transition from the core of the galaxy to the cD halo or the ICM. The radial distribution, projected ellipticity, and mean metallicity of the red (metal-richer) clusters match the halo light extremely well for , both of them varying as . By comparison, the blue (metal-poor) GC component has a much shallower falloff and a more nearly spherical distribution. This strong difference in their density distributions produces a net metallicity gradient in the GCS as a whole that is primarily generated by the population gradient. With NGC 6166 we appear to be penetrating into a regime of high enough galaxy mass and rich enough environment that the bimodal two-phase description of GC formation is no longer as clear or effective as it has been in smaller galaxies.