A search for the production of three Higgs bosons (HHH) in the bb¯bb¯bb¯ final state is presented. The search uses 126  fb-1 of proton-proton collision data at s=13  TeV collected with the ATLAS detector at the Large Hadron Collider. The analysis targets both nonresonant and resonant production of HHH. The resonant interpretations primarily consider a cascade decay topology of X→SH→HHH with masses of the new scalars X and S up to 1.5 and 1 TeV, respectively. In addition to scenarios where S is off-shell, the nonresonant interpretation includes a search for Standard Model HHH production, with limits on the trilinear and quartic Higgs self-coupling set. No evidence for HHH production is observed. An upper limit of 59 fb is set, at the 95% confidence level, on the cross section for Standard Model HHH production.