What if $\lambda_{hhh}\neq 3m_h^2/v$

A measurement of the Higgs trilinear self coupling $\lambda_{hhh}$ will test the Standard Model Higgs potential. But can it reveal information that cannot be learned otherwise? By analyzing several simple extensions of the Standard Model scalar sector we show that this measurement might give a first hint for New Physics modifying the electroweak symmetry breaking. Combining the measurements of $\lambda_{hhh}$ and $\lambda_{hVV}$ ($V=W,Z$) is particularly powerful in distinguishing between various models of New Physics and in providing unique information on these models.