Well, distributions like the Dirac delta qualify. But they can also be viewed as approximations of a continuum. Outside of that, maybe if you look somewhere li

Well, distributions like the Dirac delta qualify. But they can also be viewed as approximations of a continuum. Outside of that, maybe if you look somewhere like string theory? Still, in general, the continuum is enough. And while it takes work, I'm pretty sure that nothing relevant to physics about the continuum cannot be said within a strictly Constructivist version of the continuum. So no, there is no physical theory from which we can justify the existence of more reals than rationals. The justification must be philosophical. And the answers that we accept will strongly depend on what our notions about what existence should mean.