  We study the sensitivity to the shape of the Higgs potential of single,
double, and triple Higgs production at future $e^+e^-$ colliders. Physics
beyond the Standard Model is parameterised through the inclusion of
higher-dimensional operators $(\Phi^\dagger \Phi- v^2/2)^n/\Lambda^{(2n-4)}$
with $n=3,4$, which allows a consistent treatment of independent deviations of
the cubic and quartic self couplings beyond the tree level. We calculate the
effects induced by a modified potential up to one loop in single and double
Higgs production and at the tree level in triple Higgs production, for both $Z$
boson associated and $W$ boson fusion production mechanisms. We consider two
different scenarios. First, the dimension six operator provides the dominant
contribution (as expected, for instance, in a linear
effective-field-theory(EFT)); we find in this case that the corresponding
Wilson coefficient can be determined at $\mathcal{O}(10\%)$ accuracy by just
combining accurate measurements of single Higgs cross sections at $\sqrt{\hat
s}=$240-250 GeV and double Higgs production in $W$ boson fusion at higher
energies. Second, both operators of dimension six and eight can give effects of
similar order, i.e., independent quartic self coupling deviations are present.
Constraints on Wilson coefficients can be best tested by combining measurements
from single, double and triple Higgs production. Given that the sensitivity of
single Higgs production to the dimension eight operator is presently unknown,
we consider double and triple Higgs production and show that combining their
information colliders at higher energies will provide first coarse constraints
on the corresponding Wilson coefficient.
