  The most massive stars known to date are R 136 a1, a2, a3, and c within the
central cluster R 136a of the Tarantula nebula in the Large Magellanic Cloud
(LMC), with reported masses in excess of 150-200$M_\odot$. However, the mass
estimation of these stars relies on the assumption that they are single. We
collected three epochs of spectroscopy for R 136 a1, a2, a3, and c with the
Space Telescope Imaging Spectrograph (STIS) of the Hubble Space Telescope (HST)
in the years 2020-2021 to probe potential radial-velocity (RV) variations. We
combine these epochs with an additional HST/STIS observation taken in 2012. We
use cross-correlation to quantify the RVs, and establish constraints on
possible companions to these stars up to periods of ~10 yr. Objects are
classified as binaries when the peak-to-peak RV shifts exceed 50 km/s, and when
the RV shift is significant with respect to errors.
  R 136 a1, a2, and a3 do not satisfy the binary criteria and are thus
classified as putatively single, although formal peak-to-peak RV variability on
the level 40 km/s is noted for a3. Only R 136 c is classified as binary, in
agreement with literature. We can generally rule out massive companions (M2 >
~50 Msun) to R 136 a1, a2, and a3 out to orbital periods of < 1 yr (separations
< 5 au) at 95% confidence, or out to tens of years (separations < ~100 au) at
50% confidence. Highly eccentric binaries (e > ~0.9) or twin companions with
similar spectra could evade detection down to shorter periods (> ~10 d), though
their presence is not supported by the relative X-ray faintness of R 136 a1,
a2, and a3. We derive a preliminary orbital solution with a 17.2 d period for
the X-ray bright binary R 136 c, though more data are needed to conclusively
derive its orbit.
  Our study supports a lower bound of 150-200 $M_\odot$ on the upper-mass limit
at LMC metallicity
