A search for heavy long-lived multi-charged particles is performed using the ATLAS detector at the LHC. Data collected in 2015–2018 at s=13 TeV from pp collisions corresponding to an integrated luminosity of 139 fb−1 are examined. Particles producing anomalously high ionization, consistent with long-lived spin-½ massive particles with electric charges from |q|=2e to |q|=7e are searched for. No statistically significant evidence of such particles is observed, and 95% confidence level cross-section upper limits are calculated and interpreted as the lower mass limits for a Drell–Yan plus photon-fusion production mode. The least stringent limit, 1060 GeV, is obtained for |q|=2e particles, and the most stringent one, 1600 GeV, is for |q|=6e particles.