  We construct a resummation at partial next-to-next-to-next-to-leading
logarithmic accuracy for hadronic top-quark pair production near partonic
threshold, including simultaneously soft-gluon and Coulomb corrections, and use
this result to obtain approximate next-to-next-to-next-to-leading order
predictions for the total top-quark pair-production cross section at the LHC.
We generalize a required one-loop potential in non-relativistic QCD to the
colour-octet case and estimate the remaining unknown two-loop potentials and
three-loop anomalous dimensions. We obtain a moderate correction of $1.5\%$
relative to the next-to-next-to-leading order prediction and observe a
reduction of the perturbative uncertainty below $\pm 5\%$.
