We consider Rayleigh–Bénard convection as modelled by the Boussinesq equations in the infinite-Prandtl-number limit. We are interested in the scaling of the average upward heat transport, the Nusselt number Nu, in terms of the non-dimensionalized temperature forcing, the Rayleigh number Ra. Experiments, asymptotics and heuristics suggest that Nu ∼ Ra1/3. This work is mostly inspired by two earlier rigorous work on upper bounds of Nu in terms of Ra. (1) The work of Constantin and Doering establishing Nu ≲ Ra1/3ln 2/3Ra with help of a (logarithmically failing) maximal regularity estimate in L∞ on the level of the Stokes equation. (2) The work of Doering, Reznikoff and the first author establishing Nu ≲ Ra1/3ln 1/3Ra with help of the background field method. The paper contains two results. (1) The background field method can be slightly modified to yield Nu ≲ Ra1/3ln 1/15Ra. (2) The estimates behind the background field method can be combined with the maximal regularity in L∞ to yield Nu ≲ Ra1/3ln 1/3ln Ra — an estimate that is only a double logarithm away from the supposedly optimal scaling.