A search for the production of three Higgs bosons (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>H</a:mi><a:mi>H</a:mi><a:mi>H</a:mi></a:math>) in the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>b</c:mi><c:mover accent="true"><c:mi>b</c:mi><c:mo stretchy="false">¯</c:mo></c:mover><c:mi>b</c:mi><c:mover accent="true"><c:mi>b</c:mi><c:mo stretchy="false">¯</c:mo></c:mover><c:mi>b</c:mi><c:mover accent="true"><c:mi>b</c:mi><c:mo stretchy="false">¯</c:mo></c:mover></c:math> final state is presented. The search uses <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mn>126</k:mn><k:mtext> </k:mtext><k:mtext> </k:mtext><k:msup><k:mi>fb</k:mi><k:mrow><k:mo>−</k:mo><k:mn>1</k:mn></k:mrow></k:msup></k:math> of proton-proton collision data at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msqrt><m:mi>s</m:mi></m:msqrt><m:mo>=</m:mo><m:mn>13</m:mn><m:mtext> </m:mtext><m:mtext> </m:mtext><m:mi>TeV</m:mi></m:math> collected with the ATLAS detector at the Large Hadron Collider. The analysis targets both nonresonant and resonant production of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>H</o:mi><o:mi>H</o:mi><o:mi>H</o:mi></o:math>. The resonant interpretations primarily consider a cascade decay topology of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>X</q:mi><q:mo stretchy="false">→</q:mo><q:mi>S</q:mi><q:mi>H</q:mi><q:mo stretchy="false">→</q:mo><q:mi>H</q:mi><q:mi>H</q:mi><q:mi>H</q:mi></q:math> with masses of the new scalars <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>X</u:mi></u:math> and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>S</w:mi></w:math> up to 1.5 and 1 TeV, respectively. In addition to scenarios where <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>S</y:mi></y:math> is off-shell, the nonresonant interpretation includes a search for Standard Model <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>H</ab:mi><ab:mi>H</ab:mi><ab:mi>H</ab:mi></ab:math> production, with limits on the trilinear and quartic Higgs self-coupling set. No evidence for <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>H</cb:mi><cb:mi>H</cb:mi><cb:mi>H</cb:mi></cb:math> production is observed. An upper limit of 59 fb is set, at the 95% confidence level, on the cross section for Standard Model <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>H</eb:mi><eb:mi>H</eb:mi><eb:mi>H</eb:mi></eb:math> production. © 2025 CERN, for the ATLAS Collaboration 2025 CERN