In my first settlement, now a town, I have built up my Factory to level 13. When I click on it, there is no option to build it further. I just get the one line "Factory [13]" in the usual light brown-colored box at the top of my town screen, with a little "x" next to it.
How to continue?

Lvl 13 is the highest that you can build a factory in a City.

Later when you leave DarkAura protection and build a palace to upgrade your city to a Metropolis you will be given the option to build your Factory (and other buildings) further.

I think that the Forge is the only building that can be built to Lvl 16 in a City & Walls can be built to Lvl 75.