Specifically, I mean when we compare two related factors, such as speed and wind resistance. When we double speed, wind resistance is multiplied by four, so we have a square function described by 2y=x^2 where the exponent in question is “2” (Stop me when I’m getting this wrong).

So, my question is: What is the largest known exponent in such a relationship? The largest I remember from school involved blackbody radiation where there was a 2y=x^4, which was temperature vs radiation rate (or something like that.)

Nitpick: Both of your example equations should just say “y”, not “2y”. And there’s generally a proportionality constant of some sort in there, too. I’m guessing you put the 2 in because these things are often expressed as “when you double speed…”, or the like, but you could just as well write “when you triple speed, resistance is multiplied by nine”, or whatever.

The Lennard-Jones is the highest I know of. But, bear in mind that that is more of an empirical version of “really really really steep”. There’s no mechanistic reason to choose those exact exponents or even to choose a power law, I think.

Not sure, but I think light scattering by spheres much smaller than wavelengths has a 6 exponent in it. Not sure if there’s justification for it or if it’s empirical too.

People have wondered about why any exponent greater than about 6 doesn’t really crop up naturally. IIRC, there’s an argument about what integrals can reasonably be expected to occur in the derivations of formulae, though I equally don’t remember the argument being too convincing. Barrow and Tipler discuss the matter in The Anthropic Cosmological Principle and that’d be the place to look.

You’re thinking of Rayleigh scattering., which is proportional to the 6th power of spheres’ diameter. IIRC, it has a rigourous justification. Actually, from the same link, it looks like Rayleigh scattering occurs also in optical fibers. In this case the amount of scattered light is proportional to the 8th power of the refractive index.

It may seem like cheating, but you may be able to combine laws to get higher exponents. For example, with distance and density constant,
two bodies each of radius d attract each other gravitationally as d^6

Yes, it’s easy to get arbitrarily high exponents by looking at the right, if contrived, things. For example, if some substance decays at an exponential rate, such that after any second the amount divides by K, then, of course, after an hour, the amount divides by K[sup]3600[/sup]. One could certainly view both “decay after a second” and “decay after an hour” as reasonable measurements to concern oneself with, and this is indeed a legitimate formula relating them.

Of course, this isn’t fundamentally an empirically discovered physical law, but rather a mathematical one, concerning the phenomenon of exponential decay. Then again, many physical laws are mathematically derived from more fundamental physical laws and that’s how many of these exponents pop up anyway, so…

Slightly less contrived, perhaps, would be to construct a high-order multipole of some sort (an arrangement of charges, say). The field due to an electric monopole falls off as 1/r^2; due to a dipole, it’s 1/r^3; due to an octopole, it’s 1/r^4, and so on. Of course, in practice, you’re likely to either have the field be dominated by some low-order multipole, or to have to construct a very contrived arrangement to cancel out all of the low orders.