The production of a top-antitop quark pair in association with a W boson (tt¯W) is one of the heaviest signatures currently probed at the Large Hadron Collider. Since the first observation reported in 2015, the corresponding rates have been found to be consistently higher than the standard model predictions, which are based on next-to-leading order calculations in the QCD and electroweak interactions. We present the first next-to-next-to-leading order QCD computation of tt¯W production at hadron colliders. The calculation is exact, except for the finite part of the two-loop virtual corrections, which is estimated using two different approaches that lead to consistent results within their uncertainties. We combine the newly computed next-to-next-to-leading order QCD corrections with the complete next-to-leading order QCD plus electroweak results, thus obtaining the most advanced perturbative prediction available to date for the tt¯W inclusive cross section. The tension with the latest ATLAS and CMS results remains at the 1σ-2σ level.