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 √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.