Things like the binding energy of the electron to the hydrogen nucleus depend on C. Change C and everything changes.

You can easily imagine a world where C is very low - just speed yourself up tremendously. You end up in a world that is essentially frozen - the days are millenia long, it takes eons for anything to happen, gravity is very weak (nothing to hold you down). C represents the "stiffness" of space with respect to time - a low C means a very, very stiff world. The fact the C is large allows for a fluid existence relative to our psychological time scale. For C->infinity, gravity is impossible, and the world as we know it is impossible.

There is no free lunch. It's a trade-off - either the outer world is impossibly large or the local world is impossibly dull.