  The observed masses of the most massive stars do not surpass about 150Msun.
This may either be a fundamental upper mass limit which is defined by the
physics of massive stars and/or their formation, or it may simply reflect the
increasing sparsity of such very massive stars so that observing even
higher-mass stars becomes unlikely in the Galaxy and the Magellanic Clouds. It
is shown here that if the stellar initial mass function (IMF) is a power-law
with a Salpeter exponent (alpha=2.35) for massive stars then the richest very
young cluster R136 seen in the Large Magellanic Cloud (LMC) should contain
stars with masses larger than 750Msun. If, however, the IMF is formulated by
consistently incorporating a fundamental upper mass limit then the observed
upper mass limit is arrived at readily even if the IMF is invariant. An
explicit turn-down or cutoff of the IMF near 150Msun is not required; our
formulation of the problem contains this implicitly. We are therefore led to
conclude that a fundamental maximum stellar mass near 150Msun exists, unless
the true IMF has alpha>2.8.
