The last doubling was really significant. You know, instant free communication to humans anywhere on the world as well as instant access to the sum total of human knowledge.
Which is not to say that internet technology isn't widely more crazy than all of those previous or that there won't continue to be wider and crazier things to come. It is to say that we're not escaping exponential growth and there still isn't anything like a spike in that sort of curve.
The singularity theory proposes an increase from exponential: some technology that speeds new technology's ability to speed new technology. Without the feedback --- without the efficient feedback --- everything "stays exponential". Which is only really depressing if you're looking for hypersmart AI and endless free human life in the next 20 orso.
For example, that bit about instant access to people and knowledge was probably said about the telegraph too.