The first thing we need to look at is can we produce enough power for all 100 billion of us on Earth? This is the most important question that needs to be answered.
Power consumption
Power is a hard science factor, meaning there is no getting around it. Either we can or we can't. No amount of arm waiving and feel good motivational talks will help here.
If we assume that the world continues developing towards the current western society standards we can surely come up with a power production value per capita that we will need. But is that a reasonable starting point? Maybe not.
Developing countries have a radically lower power use per capita, but are also much less productive. So there's a political choice that needs to be made. How do we evaluate how much power we use while being productive, and how much is "waste" that we really don't need. I would argue that this actually isn't something that can be estimated. Being productive at work, and being "wasteful" in your free time, are the same thing. What we need to look at is what kind of a mix of highly skilled and paid people with a high energy consumption will we have in relation to low skill low pay low energy consumption people.
My best guess would have to be to look at countries that incorporate a wide skill and pay gap successfully, and use that as a starting point. From my estimate, the best place to look for data would then be the UAE.
Data from https://ourworldindata.org/energy/country/united-arab-emirates
That gives us about 150MWh per capita per year of energy consumption. But we use standard units here, so that equals about 17kW of power on average. Which does seem high. But they use a lot of AC, and have aluminum production etc. Maybe it's accurate. Maybe not. But this is what we'll go with for now.
So for 100 billion people, that is 1,700,000,000,000,000W or 1,700 million megawatt or 1.7 petawatts.
What are the science limits on Earths power consumption?
As with all power consumption, it all eventually goes to heat. So how much heat is the Earth capable of radiating into space? According to https://phys.org/news/2011-10-vast-amounts-solar-energy-earth.html the Earth is "hit" by 173 petawatts of power from the sun. All systems on Earth should be considered as being in a stable state (although this stable state can be changed by for example oxygen producing organisms etc) so we know the Earth can radiate at least that much heat into space. If simplifying things.
Is this a reasonable simplification though? Only approximately half the Earth is heated by the sun. And the Earth itself has significant internal heat left over from its formation, plus likely a not insignificant amount of fission happening in its core.
According to https://en.wikipedia.org/wiki/Earth's_internal_heat_budget the Earth outputs approximately 42 terawatt of heat by itself. So about 1/500 or basically a rounding error compared to the power from the sun. For weather and climate models this absolutely needs to be modeled accurately, but for our analysis this can be ignored.
1%
173 petawatt of (current) cooling power for the Earth. And 100 billion of us would consume 1.7 petawatts. 1%. Not too bad.
However we really need to consider how this energy is being produced/consumed. Are we just adding 1% to the total budget (through fossil fuels, nuclear, geothermal, off-Earth solar etc)? Or are we stealing that 1% from say solar power (reducing the suns direct effect, and outputting heat instead)?
If we can do it through adding 1% to the total heat budget we should be ok. A stable system as the Earth should not have issues with a 1% increase in total heat output. This would have significant local influence on the weather though with cities heating up causing changes to global atmosphere circulation. Basically what we have now, but more of it.
If we can't produce that amount of power ourselves (say we can't solve fusion power or off-Earth solar power for example) and end up needing to steal 1% from solar power... jikes. That would likely cause much more severe weather and climate alterations since this would have to happen primarily around the equator area. Total heat amount would of course remain the same. But the "natural" situation would be offset more. Which may or may not be a good thing. Some analysis might be required.
So can we?
100 billion people. 1.7 petawatts. Yes there's definitely science to support 100 billion people on Earth with current energy consumption levels.
If we want to be really productive though, that might be 10x? In that case we would be talking about 10% of the global heat budget. Might be ok. But very well might not be. Heavy increase in CO2 might be required to increase plant growth to offset that much extra heat if we're adding to the total heat budget. Which we would have to be since it's not feasible to be able to steal 10% of all solar power naturally hitting the Earth without killing the ecosystem. Some weather modeling required.
So if we're being very conservative, the upper limit would be about 100 billion people on Earth with current productivity. If we get very good at moving some production and development to space maybe 100 billion people with 10x the current power consumption, or 1 trillion people with current power consumption would be the upper limit. To progress further we will need to transition to a space based race.