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).