Natural laws often feature squares and square roots, and second-order differential equations. Cubic laws, cube-roots, and third-order differentials are fairly rare.
(Some counter-examples: square-cube laws turn up when area/volume effects are scaled, and Stefan–Boltzmann law involves a fourth-power. Perhaps I'm just ignorant but I struggle to come up with many more.)
Is there a deep reason why higher-order effects would be rarer?