  The star S0-2, which orbits the supermassive black hole (SMBH) in our Galaxy
with a period of 16 years, provides the strongest constraint on both the mass
of the SMBH and the distance to the Galactic center. S0-2 will soon provide the
first measurement of relativistic effects near a SMBH. We report the first
limits on the binarity of S0-2 from radial velocity monitoring, which has
implications for both understanding its origin and robustness as a probe of the
central gravitational field. With 87 radial velocity measurements, which
include 12 new observations presented, we have the data set to look for radial
velocity variations from S0-2's orbital model. Using a Lomb-Scargle analysis
and orbit fitting for potential binaries, we detect no radial velocity
variation beyond S0-2's orbital motion and do not find any significant periodic
signal. The lack of a binary companion does not currently distinguish between
different formation scenarios for S0-2. The upper limit on the mass of a
companion star ($M_{\text{comp}}$) still allowed by our results has a median
upper limit of $M_{\text{comp}}$ $\sin i \leq$ 1.6 M$_{\odot}$ for periods
between 1 and 150 days, the longest period to avoid tidal break up of the
binary. We also investigate the impact of the remaining allowed binary system
on the measurement of the relativistic redshift at S0-2's closest approach in
2018. While binary star systems are important to consider for this experiment,
we find plausible binaries for S0-2 will not alter a 5$\sigma$ detection of the
relativistic redshift.
