Population density numbers are problematic to generate or think about, especially if they are averaged over large areas. Even city population numbers are almost noise, because it all depends where you draw the borders. For some reason Tokyo often pops up as the biggest city, but you could just as easily say Los Angeles is the biggest city. It's sort of like the coastline problem.
At the block level, typical residential densities in cities are 100-300 per acre. With few exceptions, less than 100/acre is considered blight for a city, and unsustainable. That's 64,000-200,000 per square mile. In NYC, 100,000 isn't even considered a district... just a neighborhood.
There are a few buildings in Alaska that are basically an entire town in one giant building. I always thought that was pretty interesting.