Abstract Data from the James Webb Space Telescope have revealed an intriguing population of bright galaxies at high redshifts. In this work, we use extreme-value statistics to calculate the distribution (in UV magnitude) of the brightest galaxies in the redshift range 9 ≲ z ≲ 16. We combine the Generalised Extreme Value (GEV) approach with modelling of the galaxy luminosity function. We obtain predictions of the brightest galaxies for a suite of luminosity functions, including the Schechter and double power law functions, as well as a model parametrised by the stellar formation efficiency f*. We find that the JWST data is broadly consistent with f* of $5~{{\ \rm per\ cent}}-10~{{\ \rm per\ cent}}$, and that the brightest galaxy at z ∼ 16 will have $M_{\rm UV}\approx -23.5^{0.8}_{0.4}$. If f* is dependent on halo mass, we predict $M_{\rm UV}\approx -22.5^{0.5}_{1.5}$ for such an object. We show that extreme-value statistics not only predicts the magnitude of the brightest galaxies at high redshifts, but may also be able to distinguish between models of star formation in high-redshift galaxies.