A search for the exclusive hadronic decays W^{±}→π^{±}γ, W^{±}→K^{±}γ, and W^{±}→ρ^{±}γ is performed using up to 140  fb^{-1} of proton-proton collisions recorded with the ATLAS detector at a center-of-mass energy of sqrt[s]=13  TeV. If observed, these rare processes would provide a unique test bench for the quantum chromodynamics factorization formalism used to calculate cross sections at colliders. Additionally, at future colliders, these decays could offer a new way to measure the W boson mass through fully reconstructed decay products. The search results in the most stringent upper limits to date on the branching fractions B(W^{±}→π^{±}γ)<1.9×10^{-6}, B(W^{±}→K^{±}γ)<1.7×10^{-6}, B(W^{±}→ρ^{±}γ)<5.2×10^{-6} at 95% confidence level.