Natural computation based on fundamental forces Well-known examples of computation inspired by natural phenomenon are quantum computers and DNA computers.

What is known about the potential and/or limitations of computing with Maxwell's laws or gravity?

That is, incorporating nature's "quick" solutions to Maxwell's equations or the n-body problem directly into a general-purpose algorithm?
At present, quantum computation is the most powerful of computational model based on known physics to have been experimentally realised, and can efficiently simulate Maxwell's equations, and pretty much every other physical phenomenon you encounter in day to day life. As the others have mentioned, one exception to this is general spacetimes allowed as solutions in general relativity.

There has been quite a lot of interest in the computational power of computers with access to closed time like curves, for example. However there is absolutely no evidence that these exist in nature or that they can be created artificially. So, while there are potentially interesting computational models that incorporate general relativity in some form, there is significant doubt over whether such models can be realised, and before we can have the most general model of physical computation we need a solid theory of quantum gravity.

Further the interesting features of general relativity tend to only show up in regions of high curvature, which is very different from the almost flat region of spacetime we inhabit and the effects of relativity in such flat(ish) space offer no computational advantage.