A search for the production of three Higgs bosons ((Formula presented)) in the (Formula presented) final state is presented. The search uses (Formula presented) of proton-proton collision data at (Formula presented) collected with the ATLAS detector at the Large Hadron Collider. The analysis targets both nonresonant and resonant production of (Formula presented). The resonant interpretations primarily consider a cascade decay topology of (Formula presented) with masses of the new scalars (Formula presented) and (Formula presented) up to 1.5 and 1 TeV, respectively. In addition to scenarios where (Formula presented) is off-shell, the nonresonant interpretation includes a search for Standard Model (Formula presented) production, with limits on the trilinear and quartic Higgs self-coupling set. No evidence for (Formula presented) production is observed. An upper limit of 59 fb is set, at the 95% confidence level, on the cross section for Standard Model (Formula presented) production.