We study higher-order QCD corrections for the associated production of a top-antitop quark pair and a W boson ($t\overline{t}W$ production) in proton-proton collisions. We calculate approximate NNLO (aNNLO) and approximate N$^{3}$LO (aN$^{3}$LO) cross sections, with second-order and third-order soft-gluon corrections added to the exact NLO QCD result, and we also include electroweak (EW) corrections through NLO. We calculate uncertainties from scale dependence, which are reduced at higher orders, and from parton distributions, and we also provide separate results for $t\overline{t}{W}^{+}$ and $t\overline{t}{W}^{-}$ production. We compare our results to recent measurements from the LHC, and we find that the aN$^{3}$LO QCD + NLO EW predictions provide improved agreement with the data. We also calculate differential distributions in top-quark transverse momentum and rapidity and find significant enhancements from the higher-order corrections. This is the first calculation of soft-gluon corrections in single-particle-inclusive kinematics for a $2\to 3$ process where all final-state particles are massive.