A search for doubly-charged Higgs bosons decaying to pairs of same-charge, isolated, high-pT electrons is presented. Same-charge electron pairs are selected in an inclusive way and the same-charge electron pair invariant mass spectrum is studied. The search uses the $pp$ collision data sample corresponding to 13.9\,\mathrm{fb}^{-1} of integrated luminosity collected in 2015 and 2016 by the ATLAS detector at the LHC at the centre-of-mass energy of 13\,\mathrm{TeV}. A model independent search for new physics processes was performed and no clear evidence for an excess was observed. Therefore, the upper limits for the signal cross-sections were set at 95% confidence level. Consequently, the observed lower mass limits, assuming a $100\%$ branching ratio to electrons ($\mathrm{Br}(H^{\pm \pm}\rightarrow e^{\pm}e^{\pm})=100\%$), are 420\,\mathrm{GeV} for a doubly-charged Higgs boson coupling to right-handed leptons ($H_R^{\pm \pm}$) and 570\,\mathrm{GeV} for a doubly-charged Higgs boson coupling to left-handed leptons ($H_L^{\pm \pm}$). The expected lower mass limits were 460\,\mathrm{GeV} for $H_R^{\pm \pm}$ and 580\,\mathrm{GeV} for $H_L^{\pm \pm}$. Limits were also derived with the assumption that $\mathrm{Br}(H^{\pm \pm}\rightarrow e^{\pm}e^{\pm})=50\%$. In this case, the lower mass limits are 380\,\mathrm{GeV} for $H_R^{\pm \pm}$ and 530\,\mathrm{GeV} for $H_L^{\pm \pm}$. The expected limits for this assumption were 400\,\mathrm{GeV} for $H_R^{\pm \pm}$ and 530\,\mathrm{GeV} for $H_L^{\pm \pm}$.

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