10 billion. Hans Rosling makes a compelling case from the numbers that the world population will peak at 10 billion, then slowly decline, because we've already passed "peak child" the year in which the largest number of children were born, that number is now gradually declining. Population will continue to increase for a while because the older cohorts are currently much smaller than the younger cohorts, so as the younger cohorts age into the older categories, we'll have a "filling out" of the age distribution.
However, I don't think there's any reason to assume that we're going to stagnate. We're also on the cusp of a set of new technologies which will make the human race dramatically wealthier, because we've barely scratched the surface of the potential of information technology and automation.
The only fundamental limitation to our growth is energy production and delivery. We're heading towards a day when we can no longer rely on our current primary energy source, fossil fuels, perhaps because we'll exhaust the easily-reached reserves, and perhaps because we'll decide that they're too difficult to use without excessive environmental impact. But we're also rapidly improving our ability to capture solar energy, in its various forms. For that matter, we've successfully performed net-positive controlled fusion; perhaps we'll learn to harness that.
I do think the west, especially the US, is likely headed for a period of slower growth than we're accustomed to, or perhaps worse, stagnation or decline. This is because globalization (which many think is a dirty word, but I think is fantastic) is spreading the wealth over more of the human race.
This may seem to contradict the other current trend of concentration of capital, but historically they've gone hand in hand. During the initial expansion phase massive fortunes are created as the masses reap the benefits of the new capabilities in their personal lives, but the bulk of the financial gain goes to a relative few. Then, as technology matures the productivity gains spread; the fortunes don't disappear, but new fortunes are created at a slower rate and the wealth gap closes, because competition drives out the massive profit margins leaving the wealth in the hands of more people.
For that matter, I'm far from convinced there isn't another technological revolution on the horizon, and another after that, and so on. It's tempting to believe that the knowledge we've achieved so far is nearing the limit, but very bright people of generations and centuries past have believed the same thing... and they've all been not just wrong, but stupefyingly wrong. In fact, there seems to be a strong correlation between the number of very smart people who are convinced that all that's left to discover is making what we already know more precise and the closeness of the next earth-shattering discovery.
Indeed, we have powerful reasons right now to believe that we're on the cusp of yet another revolution in our understanding of physics. We currently have two thoroughly-elaborated and extensively supported models of the structure of reality... and they're mutually contradictory. Relevant to the current article, our understanding of biology is advancing at a breakneck pace, as we move from understanding only gross biological processes to elaborating the detailed chemical processes and, even more important, how they're described, defined and implemented by genetic material. Informatics is crucial to that effort.
I can't think of a single field of research in which we aren't currently learning orders of magnitude faster than ever before, and which hasn't recently seen, or appears to be in a position to soon find, revolutionary new understanding. In that context how can we not expect additional technological revolutions? But even without new fundamental science, I think it's clear that we've barely scratched the surface of what information technology can do for us. There's plenty of growth ahead, and the population problem appears to solve itself neatly.
Which isn't to say there aren't challenges coming. There are, and we can even see what some of them are. But this post is already long enough.