How big can a black hole grow?

I show that there is a physical limit to the mass of a black hole, above which it cannot grow through luminous accretion of gas, and so cannot appear as a quasar or active galactic nucleus (AGN). The limit is M-max similar or equal to 5 x 10(10)M(circle dot) for typical parameters, but can reach M-max similar or equal to 2.7 x 10(11) M-circle dot in extreme cases (e. g. maximal prograde spin). The largest black hole masses so far found are close to but below the limit. The Eddington luminosity similar or equal to 6.5 x 10(48) erg s(-1) corresponding to Mmax is remarkably close to the largest AGN bolometric luminosity so far observed. The mass and luminosity limits both rely on a reasonable but currently untestable hypothesis about AGN disc formation, so future observations of extreme supermassive black holemasses can therefore probe fundamental disc physics. Black holes can in principle grow their masses above Mmax by non-luminous means such as mergers with other holes, but cannot become luminous accretors again. They might nevertheless be detectable in other ways, for example through gravitational lensing. I show further that black holes with masses similar to M-max can probably grow above the values specified by the black-hole-host-galaxy scaling relations, in agreement with observation.