Berlin has 4,227 inhabitants/km², Hong Kong (widely considered to be a very livable city, and also one with hard geographic constraints on expansion) has 6,300. So we have existence proof that Berlin, if it chose to, could support an increase in housing of 50% or so, which would certainly drive rents way down.
Edit: Not sure why I looked so far away, Paris is at 21,000. Like virtually every housing story, this is an example of the people who live in a place deciding they don't want anyone else to live there, and putting up economic walls that only the rich can surmount.
> Berlin has 4,227 inhabitants/km² [...] Paris is at 21,000.
These numbers aren't comparable: The municipality of Paris is just the small urban core of a much larger metropolitan region. Berlin has about 3.7 million inhabitants on 890 square kilometers, Paris 2.2 million on 105 square kilometers.
A fairer comparison is with the Petite Couronne consisting of Paris + the innermost ring of its neighbors (https://en.wikipedia.org/wiki/%C3%8Ele-de-France#Petite_Cour...), which has 6.7 million people on 760 square kilometers. Still denser than Berlin! But the ratio is far less extreme than you made it look. For an alternative approach, take Berlin's densest districts: Friedrichshain-Kreuzberg, Mitte, and Neukölln add up to about 100 square kilometers (like Paris) and a bit over a million people. So about 2x less dense than Paris rather than 5x.
Edit: Not sure why I looked so far away, Paris is at 21,000. Like virtually every housing story, this is an example of the people who live in a place deciding they don't want anyone else to live there, and putting up economic walls that only the rich can surmount.