?? Centuries? You mean twenty/thirty years, probably? We don't really have computers until about 50 years ago, and even then it was not widespread.
You don't have to wonder what post-Moore's law will look like. Look at other industries when they become mature. The car industry is one example: there's not that much innovation that goes into making a car better in terms of performance. I mean, a car from 30 years ago may be less comfortable but still drives pretty well as long as it's maintained. What makes the difference then is not performance anymore but whatever added value you bring on it: durability, reliability, service, etc... no doubt sooner or later electronics makers will have to innovate in different directions as well.
And while I've been watching promises of quantum computing for a decade or more with no apparent real progress, it's possible that alternative models of computing devices will allow still greater computational power.
I'd also expect more of EC2 and GCE to happen, and ironically the "small" developer to pay less, but also have less options.
The past couple centuries has been dominated by consumers and businesses automating and digitizing increasing parts of their lives as computers become more capable to handle them at the same low cost. (And the power the Internet gives us for being able to share information)
As creating faster chips becomes harder, institutions will start to invest more in other promising fields and technologies. I wonder if the next few decades will be dominated with developments from an entirely new discipline and culture.