If we increase our energy consumption by 1 % per year, we will all be dead in less than 1000 years because at that point the surface temperature of Earth will have reached the boiling point of water because it can not radiate the energy into space. [1]
You may object that we can reduce the energy consumption per capita and therefore have population growth without an increased energy consumption but that does not work.
Starting with the population of 7.4 billion people in 2016 and a population growth rate of 1 % per year, it will take less than 1400 years until body heat alone will raise the surface temperature of Earth above the boiling point of water. [2]
You may now object that we will of course move into space and leave Earth instead of sticking around here until we all get boiled alive. Unfortunately this does not help much either.
With a population growth rate of 1 % per year we have 2685 more years until we run out of places to live in the Milky Way assuming 400 billion stars and one Earth-like colonizable planet per star. Unfortunately 2685 years is not nearly enough time to reach all the stars in the Milky Way. Even if we would be traveling at the speed of light, we could cross just about 2 % in that time.
Even more fundamentally, space has three dimensions and therefore the number of stars and planets we can reach only increases with the third power assuming a homogeneous distribution [3] and will in consequence quickly be overwhelmed by exponential population growth within the reachable volume.
We can push all those numbers to tens of millennia with a growth rate of 0.1 % per year, but that is still the blink of an eye in the grand scheme of things. So there are very real limits to the possible growth of the human population and the time frames are not that large even with moderate growth rates. [4]
[1] The total emission of an ideal black body with surface area of 510.1 million km² at 373.15 K is 560,791 TW. The world total primary energy supply in 2012 is estimated to have been 155,505 TWh which is 17.75 TW on average. This is a factor of 31,591 from what Earth could radiate into space at 100 °C or 1,041 years of 1 % growth per year.
[2] Assuming an average basal metabolic rate of 80 W, a population growth rate of 1 % and an initial population of 7.4 billion people, it takes 1,383 years to reach an body heat output of 560,791 TW.
[3] Within a galaxy like the Milky Way it actually becomes more like the second power after some time due to the flat shape.
[4] All this assuming a form of living not to different from today. Moving human brains into computers and getting rid of the bodies would certainly impact the numbers, but even then computers will still consume energy. Similarly the numbers could be impacted by major revolutions in physics. I mean really, really major changes.