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my understanding used to be that minor characters get generated automatically over the course of the game, so you always have people around to adopt and give offices to. Lately I'm not sure of that anymore, I've reached the point in my Rome campaign were I don't have enough characters to serve as commanders for all my legions, offices, governors and researchers.
I know they get auto generated if you don't have enough for your major families but I don't know if it takes into account open jobs.
 
I haven’t hit a cap on total characters, but it seems (as monarchy) that I hit a “birth cap”. I could marry my entire male family to lustful spouses, yet not have a new birth until some other character dies off. I don’t really want to adopt, but I can’t ever seem to grow the family in a normal sense. I wonder if it’s due to being large (800+ territories) with lots of integrated cultures. I have a constant and uncontrollable flow of minor characters coming in from all the different cultures and it seems to cause births to stagnate.