  Recent data from the James Webb Space Telescope suggest that there are
realistic prospects for detecting the earliest generation of stars at redshift
~20. These metal-poor, gaseous Population III (Pop III) stars are likely in the
mass range 10-1000 solar masses. We develop a framework for calculating the
abundances of Pop III stars as well as the distribution of the most massive Pop
III stars based on an application of extreme-value statistics. Our calculations
use the star formation rate density from a recent simulation to calibrate the
star-formation efficiency from which the Pop III stellar abundances are
derived. Our extreme-value modelling suggests that the most massive Pop III
stars at redshifts 10-20 are likely to be $\gtrsim10^3-10^4\,{\rm M}_\odot$.
Such extreme Pop III stars were sufficiently numerous to be the seeds of
supermassive black holes at high redshifts and possibly source detectable
gravitational waves. We conclude that the extreme-value formalism provides an
effective way to constrain the stellar initial mass function.
