Correct me if I’m wrong here – Oldtown supposedly has ~500k people, KL is the same order of magnitude but slightly bigger (maybe 550k). AWOIAF says that Valasyr, Mantarys, and Volon Therys are bigger population wise than any of the cities in Westeros, meaning that Volantis must be huge. What’s your best guess for the population of each of the 9 free cities (or at least just the ones we’re semi-familiar with – Braavos, Pentos, and Voltanis)?

You’re quite right. And this is a very interesting question.

The historical Constantinople at its height was around 1 million, so Volantis is probably at least that big. We learn from Victarion that their war fleet is somewhere between 300-500 war galleys, which means at least 30,000 and as high as 150,000 men manning the oars (depending on whether these are 100 oarsmen or 300 oarsmen ships), not including the marines and the regular troops. Using our normal 1% rule, that gives us a population under Volantis’ control (since they have colonies, not just the metropolitan area itself) of anywhere from 3 million to 15 million. It’s definitely the largest, which makes sense as it’s the oldest and thus has had time to build its population over time.

On the other hand, only 500k to 2.5 million of their people are Volantene citizens. The rest are slaves. 

Braavos is a bit harder to estimate. Historical Venice and Amsterdam in their Golden Ages had around 200,000 residents. On the other hand, historical Venice only had 6 islands and the Mestre, whereas the metropolitan area of Braavos has 100 islands to build on. It’s highly crude, but if we simply multiply Venice’s population by 16.6, we get a population of 3.3 million. Taking another method, Braavos has the largest war fleet in the world, so it has to have substantially more than Volantis’ fleet. If it’s 600, that suggests a population of anywhere from 6 million to 18 million. 

18 million seems high, there’s no way Braavos is half the size of Westeros in terms of population even with its colonies

. Ditto Volantis, even with its colonies. So I’m more inclined to say that Volantis probably has around 300 ships, and a max population of 9 million people. Braavos probably has at least 400 ships, and a max population of 12 million people. 

The Three Daughters (Lys, Myr, and Tyrosh) have at least 90 warships, which means a combined population of between 900,000 and 2.7 million.