The observed masses of the most massive stars do not surpass about 150 Msolar. 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 the observation of 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 (α= 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 750 Msolar. 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 down-turn or cut-off of the IMF near 150 Msolar is not required: our formulation of the problem contains this implicitly. We are therefore led to conclude that a fundamental maximum stellar mass near 150 Msolar exists, unless the true IMF has α > 2.8.