The observed masses of the most massive stars do not surpass about 150 M _ { \odot } .
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 750 M _ { \odot } .
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 150 M _ { \odot } 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 M _ { \odot } exists , unless the true IMF has \alpha > 2.8 .