We're closer than most people think. Isaac Asimov's essay "The Power of Progression" from his monthly science column in the May 1960 issue of "The Magazine of Fantasy and Science Fiction", and reprinted in the collect "The Stars in Their Courses" in 1971 [1], goes through the math for how long it would take to reach various population levels at the rate the population was growing at the time.
It was doubling every 47 years then. That's slowed down to about every 63 years currently, so the times to reach those levels would now be longer, but his original numbers are still instructive for how fast exponential growth can get away from you. Here they are.
• Suppose we populated the entire Earth to the density of Manhattan at noon on a typical working day, when it is at its highest density. By "entire Earth" that means covering every square meter of the surface, including the oceans, with people.
That would take 585 years.
• Suppose that every single star in our galaxy had 10 habitable planets with the same area as Earth, and that we could trivially expand to them. And suppose that there are another hundred billion galaxies with the same number of habitable planets as ours, that we can also expand to. Surely now we are set for billions of years, right?
Nope. That just takes 4200 years at the growth rate he used (doubling every 47 years).
• OK, suppose we don't limit ourselves to planets. We find some way to tap into hyperspace or something for energy, freeing us to use all the mass in the universe for human bodies. All the stars in all the galaxies converted to humans. Let's multiply the available mass by 100 to account for dust and debris and other interstellar and intergalactic matter.
It takes 6700 years to reach that level.
• He also looks at how long, given a population that consists just of humans and plants for the humans to eat, limited only by the amount of solar energy and the efficiency of photosynthesis, until we are at the limit. That's 624 years.
I've also seen a similar calculation that uses the volume of an average human and the speed of light. I had thought it was in that Asimov essay, but it isn't. That one asks when, if you were to take every human and pack them into the smallest sphere that could them all, would the radius of that sphere have to be increasing faster than the speed of light to accommodate population growth. I don't remember the answer, but I think it was somewhere in the several thousand years range.
There's no amount of even remotely plausible technological hand waving one can do to make a case that we could ever actually achieve any of the above, except maybe the "cover the Earth" one.
To convert the numbers above to other doubling intervals, just multiply by the doubling interval you want to use, and divide by the one Asimov used (47 years).
>That's slowed down to about every 63 years currently
The global fertility rate is 2.426, which is barely above the replacement rate. The population continues to grow primarily because the people who have already been born have good life expectancies. The UN forecasts that the population will reach a maximum of 10-12 billion by 2100, driven mainly by increasing life expectancy in the developed world. Key quote:
Two-thirds of the projected growth of the global population through 2050 will be driven by current age structures. It would occur even if childbearing in high-fertility countries today were to fall immediately to around two births per woman over a lifetime.
See the sibling comment by jdietrich.
And if you are still not convinced, this short video: https://www.gapminder.org/answers/how-reliable-is-the-world-... and, if you're interested more on why the forecast is so flat, I highly recommend taking the test at gapminder.org then watching Hans' TED talks.
I mean, a bunch of O’Neill cylinders turns the carrying capacity of our solar system into potentially trillions of people if not more. Turning the entirety of habitable ground into Manhattan is hilariously far off just because we can expand habitable ground pretty damned far.
If we can make enough O'Neill cylinders to handle a trillion (10^15) people, that would buy 440 years at the current grown rate (doubling every 63 years) starting from the current population (7.7 billion).
If we can build enough to handle 1000 times that population, one quadrillion people, that can handle 1000 years of growth.
An upper limit on the number of people we can have without leaving the solar system is about 28 octillion (2.8 x 10^28), obtained by dividing the mass of the sum (1.9891 x 10^30 kg) by the mass of an average human (70 kg).
At the current growth rate, that's 3900 years away.
Indefinite exponential growth can overwhelm pretty much anything.