Here's one I like.
A human diet is (very roughly) 2000 calories per day - this converts out to (really roughy now, but within napkin range) an average power of 0.1W per person.
The earth (atmosphere, land, and water) absorbs about 3.85E24 Joules/year of solar energy (lots gets reflected away). That's just over 1E17 Watts, let us call it 1.2E17 W
That would mean that the largest sustainable human population, stacked up Matrix style and converting 100% of absorbed incoming solar energy directly into sustaining human biological function, would be 1.2E18 (about half a billion times as many people as there are now). Since the people are stacked Matrix-style they probably don't need 2000 calories per day so you might be able to double that number, lets call it 2.4E18 (this also makes the division in the next step trivial lol).
However, the surface area of the earth (land and water added) is on the order of 5.1E14 square metres. That would mean that you'd have 4000 people per square meter - not very comfortable.
The city today with the highest population density is Manila, with 4.3E4/square km (or 4.3E-2/square metre - inverted means something like 25 square metres per person).
At that density, utilizing land and sea (and mountains and Antarctica), you're looking at a maximum population of 2E13 ("only" about 3000 times the current population). To maintain their biological function you'd only need to attain on the order of 0.001% efficiency from the incoming solar energy. That leaves lots of room for other energy requirements such as computers and internets.
So I'm gonna say that, barring either extra-terrestrial energy sources or Matrix-style stacking, and assuming that just 1/10 of the planet's surface is actually hospitable for life (taking away oceans, mountains, deserts, and polar regions) you can make a good stab at determining the largest truly sustainable, long-term human population on eart.
With the collected assumptions of:
-The ability to utilize at least 0.001% of the total incoming solar energy from the sun to the earth
-A 2000 calorie diet for human beings
-A population density similar to Manila's today
-Roughly 1/10 of the earth's surface suitable for habitation
The final estimate is on the order of 2E12/two trillion/2,000,000,000,000.