  We present a spectroscopic analysis of the most rapidly rotating stars
currently known, VFTS 102 ($v_{e} \sin i = 649 \pm 52$ km s$^{-1}$; O9: Vnnne+)
and VFTS 285 ($v_{e} \sin i = 610 \pm 41$ km s$^{-1}$; O7.5: Vnnn), both
members of the 30 Dor complex in the Large Magellanic Cloud. This study is
based on high resolution ultraviolet spectra from HST/COS and optical spectra
from VLT X-shooter plus archival VLT GIRAFFE spectra. We utilize numerical
simulations of their photospheres, rotationally distorted shape, and gravity
darkening to calculate model spectral line profiles and predicted monochromatic
absolute fluxes. We use a guided grid search to investigate parameters that
yield best fits for the observed features and fluxes. These fits produce
estimates of the physical parameters for these stars (plus a Galactic
counterpart, $\zeta$ Oph) including the equatorial rotational velocity,
inclination, radius, mass, gravity, temperature, and reddening. We find that
both stars appear to be radial velocity constant. VFTS 102 is rotating at
critical velocity, has a modest He enrichment, and appears to share the motion
of the nearby OB association LH 99. These properties suggest that the star was
spun up through a close binary merger. VFTS 285 is rotating at $95\%$ of
critical velocity, has a strong He enrichment, and is moving away from the R136
cluster at the center of 30 Dor. It is mostly likely a runaway star ejected by
a supernova explosion that released the components of the natal binary system.
