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ABSTRACT The hunt is on for dozens of protoplanets hypothesized to reside in protoplanetary discs with imaged gaps. How bright these planets are, and what they will grow to become, depend on their accretion rates, which may be in the runaway regime. Using 3D global simulations, we calculate maximum gas accretion rates for planet masses Mp from 1$$\, \mathrm{ M}_{{\oplus }}$$ to $$10\, \mathrm{ M}_{\rm J}$$. When the planet is small enough that its sphere of influence is fully embedded in the disc, with a Bondi radius rBondi smaller than the disc’s scale height Hp – such planets have thermal mass parameters qth ≡ (Mp/M⋆)/(Hp/Rp)3 ≲ 0.3, for host stellar mass M⋆ and orbital radius Rp – the maximum accretion rate follows a Bondi scaling, with $$\max \dot{M}_{\rm p} \propto \rho _{\rm g}M_{\rm p}^2 / (H_{\rm p}/R_{\rm p})^3$$ for ambient disc density ρg. For more massive planets with 0.3 ≲ qth ≲ 10, the Hill sphere replaces the Bondi sphere as the gravitational sphere of influence, and $$\max \dot{M}_{\rm p} \propto \rho _{\rm g}M_{\rm p}^1$$, with no dependence on Hp/Rp. In the strongly superthermal limit when qth ≳ 10, the Hill sphere pops well out of the disc, and $$\max \dot{M}_{\rm p} \propto \rho _{\rm g}M_{\rm p}^{2/3} (H_{\rm p}/R_{\rm p})^1$$. Applied to the two confirmed protoplanets PDS 70b and c, our numerically calibrated maximum accretion rates imply that their Jupiter-like masses may increase by up to a factor of ∼2 before their parent disc dissipates.