Nonlinearities in quantum mechanics? Linearity is what makes quantum mechanics amplitude-independent. If you ruin that, then the laws of nature will change from moment to moment as the wavefunction moves to fill more and more of Fock space. Suffice it to say, QM's leading order is 1, and any higher powers are way out of reach.

Unless, that is, worlds are top-level entities in your physical theory somehow, which then brings in the full weight of the 'what does it have to do, kill a puppy' rant against it.