Exponential growth, even with a small exponent, is astounding.
Suppose, for instance, we squeeze all of humanity into one densely populated volume. Let's give each person just one cubic meter. As population grows, we expand that volume, maintaining the density of one person per cubic meter. With all of space at our disposal, we are now set for a very long time, right?
Right, we are, but probably not as long as most people would guess. Let's say that the humans of Densopolis have a growth rate of 0.01%. Each year the population is 1.0001 times the year before.
After 900k years of that growth, the frontier of Densopolis needs to be expanding faster than the speed of light to make room.
If they grow at 0.1%, or 1.001x per year, it only takes 84k years to hit the speed of light limit. At 1%, or 1.01x per year, they hit it in under 8k years.
Another neat illustration. Instead of packing densely, let's say humanity spreads throughout the Milky Way. If they maintain a growth rate of 1.0001x per year, after 940k years all the mass of the Milky Way will have been used to make human bodies. There will be none left for stars, interstellar dust, planets, or any other life forms.
At 1.001x, we need all the mass after 94k years. 1.01x does it in under 10k years.