Most processes involving people (and lots of others, besides) that look exponential at some point are really logistic, so even in that case, there's probably some equilibrium point somewhere in the future, though its possible that its many generations down the line from the point where the appearance of exponential change first occurs.
On the other hand, I'd like to say that it's also possible that some particular processes can follow exponential laws without any restriction. It might be somewhat hazy, but what if outside of a few measures (infant mortality, life expectancy, etc) economic progress is nothing more than a subjective measurement of how present consumers value past output?
I'm not saying anything new here, both Easterlin's paradox and the idea of a "hedonic treadmill" support the idea that more income doesn't necessarily equate into increased happiness. And specially in the case of economic innovation, which might not actually be technically interesting or require any "new physics" at all - just mashing some current technologies into a marketable product - it might never ever stop, and will just degenerate into an endless chase of ever changing fads. Past economies wouldn't be particular poor, just particularly quaint, and a repository of dead fads.
There doesn't need to be anything "fundamentally different" except that we finally reached the exponential part of technological progress, which up until now seemed like a linear process. Just like population growth until 1800s, or economic growth until the 1820s. Those two were pretty much linear and not very different from zero until they weren't.
I don't actually subscribe to this rationale, though. I'm just putting it out like this because it's pretty much what everybody says and I recognize that it's quite plausible, given the two first hypotheses.