Multi-Higgs Production allows us to probe the Higgs self-couplings. Due to the dominance of the Higgs-to-top couplings, those processes are primarily mediated by gluon fusion, which at leading order are already loop-induced. Triple Higgs production exhibits cross sections on the order of attobarns and is thus beyond the reach of the LHC. Nevertheless, sensitivity studies have been initiated by experimental collaborations, and the process has gained interest from the theory community, particularly in the context of physics beyond the Standard Model. Computations of two-loop processes for 2→3 kinematics with massive loops are at the forefront of what can currently be achieved in the field of precision calculations. Triple Higgs production serves as a case study for testing new analytic or semi-analytic methods that can be applied to the computation of NLO QCD corrections. In this thesis, we calculate the exact leading-order cross section of Triple Higgs production at the LHC. Following this, we apply the transverse momentum expansion technique for the first time to a 2→3 process in order to reduce the amount of scale and verify that it correctly reproduces the exact behavior within its region of validity.