  We study the production of $t{\bar t}h$ and $t{\bar t}hh$ at hadron
colliders, in the minimal Composite Higgs Models, based on the coset
$SO(5)/SO(4)$. We explore the fermionic representations ${\bf 5}$ and ${\bf
14}$. A detailed phenomenological analysis is performed, covering the energy
range of the LHC and its High Luminosity upgrade, as well as that of a future
100 TeV hadron collider. Both resonant and non-resonant production are
considered, stressing the interplay and complementary interest of these
channels with each other and double Higgs production. We provide sets of
representative points with detailed experimental outcomes in terms of
modification of the cross sections as well as resonance masses and branching
ratios. For non-resonant production, we gauge the relative importance of
Yukawa, Higgs trilinear, and contact $t\bar{t}hh$ vertices to these processes,
and consider the prospect for distinguishing the fermion representations from
each other and from the Standard Model. In the production of top partners, we
find that the three-body decay channel $W^+ W^- t$ becomes significant in
certain regions of parameter space having a degenerate spectrum, and is further
enhanced with energy. This motivates both higher energy machines as well as the
need to go beyond the current analysis performed for the searches for these
resonances.
