Physicists are developing an alternative to special relativity, called Doubly Special Relativity. (When the Wikipedia article for this first appeared, people were so sure that the name had to be a joke that the article was nominated for deletion.) Doubly special relativity, which currently is somewhere in between an idea and an actual full-fledged theory, depends on two parameters, the speed of light and the Planck mass (this puts the two in “doubly”). For an introduction, see Introduction to Doubly Special Relativity by Jerzy Kowalski-Glikman.