A search for the production of three Higgs bosons (𝐻⁡𝐻⁡𝐻) in the 𝑏⁢
¯
𝑏
⁢𝑏⁢
¯
𝑏
⁢𝑏⁢
¯
𝑏
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 𝐻⁡𝐻⁡𝐻. The resonant interpretations primarily consider a cascade decay topology of 𝑋 →𝑆⁢𝐻 →𝐻⁡𝐻⁡𝐻 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 𝐻⁡𝐻⁡𝐻 production, with limits on the trilinear and quartic Higgs self-coupling set. No evidence for 𝐻⁡𝐻⁡𝐻 production is observed. An upper limit of 59 fb is set, at the 95% confidence level, on the cross section for Standard Model 𝐻⁡𝐻⁡𝐻 production.