The first measurement of the inclusive cross section for top quark pairs ((t\stackrel{‾}{t})) produced in association with two additional charm jets is presented. The analysis uses the dileptonic final states of (t\stackrel{‾}{t}) events produced in proton-proton collisions at a centre-of-mass energy of 13 TeV. The data correspond to an integrated luminosity of (41.5\phantom{\rule{0.25em}{0ex}}{\text{fb}}^{-1}), recorded by the CMS experiment at the LHC. A new charm jet identification algorithm provides input to a neural network that is trained to distinguish among (t\stackrel{‾}{t}) events with two additional charm ((t\stackrel{‾}{t}c\stackrel{‾}{c})), bottom ((t\stackrel{‾}{t}b\stackrel{‾}{b})), and light-flavour or gluon ((t\stackrel{‾}{t}\text{LL})) jets. By means of a template fitting procedure, the inclusive (t\stackrel{‾}{t}c\stackrel{‾}{c}), (t\stackrel{‾}{t}b\stackrel{‾}{b}), and (t\stackrel{‾}{t}\text{LL}) cross sections are simultaneously measured, together with their ratios to the inclusive (t\stackrel{‾}{t}) + two jets cross section. This provides measurements of the (t\stackrel{‾}{t}c\stackrel{‾}{c}) and (t\stackrel{‾}{t}b\stackrel{‾}{b}) cross sections of (10.1±1.2\phantom{\rule{0.25em}{0ex}}\text{(stat)}±1.4\phantom{\rule{0.25em}{0ex}}\text{(syst)}\phantom{\rule{0.2em}{0ex}}\text{pb}) and (4.54±0.35\phantom{\rule{0.25em}{0ex}}\text{(stat)}±0.56\phantom{\rule{0.25em}{0ex}}\text{(syst)}\phantom{\rule{0.2em}{0ex}}\text{pb}), respectively, in the full phase space. The results are compared and found to be consistent with predictions from two different matrix element generators with next-to-leading order accuracy in quantum chromodynamics, interfaced with a parton shower simulation.