[scrabblehack]

Does the special theory actually cite a number like that, or does it leave everything in terms of c?

[FredZarguna]

No, the actual value is one which would have been of little or no interest to Einstein [or any other theoretician] except to the extent that it's being "very large" [in some sense] explains why we were unaware of relativity for so long.

Here's one of the very first serious physicists' take on the matter:

"The speed of light, if not instantaneous, is extraordinarily rapid." -- Galileo

The symbol "c" itself was not used in Einstein's paper on Special Relativity. He used the symbol "v."

[Moonman62]

The speed of light is actually dependent on two other constants which describe the behavior of electricity and magnetism in a vacuum. The speed of light and most of the concepts of relativity were already known when Einstein wrote his paper. His great genius was due to two postulates:

From Wiki.

First, he applies the principle of relativity, which states that the laws of physics remain the same for any non-accelerating frame of reference (called an inertial reference frame), to the laws of electrodynamics and optics as well as mechanics. In the second postulate, Einstein proposes that the speed of light has the same value in all inertial frames of reference, independent of the state of motion of the emitting body.