  In this work we analyse the triple Higgs boson production cross section at
hadron colliders via gluon fusion, and present for the first time the full set
of QCD NNLO corrections in the heavy top limit. In order to account for finite
top mass effects we perform two different reweighting procedures, and study the
dependence of the result on the choice of the approximation. Combining the most
accurate predictions available to date, we present the following result for the
total NNLO cross section for triple Higgs boson production at a 100 TeV
collider: $\sigma_\text{NNLO}=5.56^{+5\%}_{-6\%} \pm 20\%$ fb, where the first
uncertainty is an estimate for higher order effects from scale variations,
while the last one is an estimate for the missing finite top mass effects.
