Post-Carbon Economy -- A Thought

Has there ever been any kind of economy not based on carbon?

I think the first time I asked this question on Twitter I added a "non-trivial" qualifier to it, but the more I think about it the less I think the question needs qualification.

And it isn't just energy -- there is a gigantic amount of plastic and other polymers in the environment. I'm looking at my desk and I don't think I can find anything on it that didn't involve carbon in its makeup or creation.