A search for heavy long-lived multicharged particles is performed using the ATLAS detector at the LHC. Data with an integrated luminosity of 36.1 fb-1 collected in 2015 and 2016 from proton-proton collisions at s=13 TeV are examined. Particles producing anomalously high ionization, consistent with long-lived massive particles with electric charges from |q|=2e to |q|=7e, are searched for. No events are observed, and 95% confidence level cross-section upper limits are interpreted as lower mass limits for a Drell-Yan production model. Multicharged particles with masses between 50 and 980-1220 GeV (depending on their electric charge) are excluded.