This writeup presents the first observation of the simultaneous production of three $J/\psi$ mesons in proton-proton (pp) collisions, based on a data sample collected by the CMS experiment at a center-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 133 fb$^{-1}$. The $pp \to J/\psi\,J/\psi\,J/\psi\, X$ process is observed with a significance above five standard deviations in final states with three $\mu^{+}\mu^{-}$ pairs. The measured inclusive fiducial cross section of $\sigma(pp \to J/\psi\,J/\psi\,J/\psi\,X)= 272^{+141}_{-104} \text{(stat)} \pm 17 \text{(syst)}$ fb is compared to theoretical expectations for the production of three $J/\psi$ mesons in single- (SPS), double- (DPS), and triple- (TPS) parton scatterings. Assuming factorization of multiple hard-scattering probabilities in terms of SPS cross sections, the measured process is found to be dominated by DPS and TPS contributions, and an effective DPS cross section, related to the transverse distribution of partons in the proton, of $\sigma_{eff,DPS} = 2.7^{+1.4}_{-1.0} \text{(exp)} ^{+1.5}_{-1.0} \text{(theo)}$ mb is determined.