The rules for short stays (with or without visa) are already explained in How does the Schengen 90/180 rule work? and extending your stay is simply not possible. Legally, Schengen member states can still issue something called a “limited territoriality visa” to allow you to stay but this is intended for diplomats or humanitarian purposes so it won't be available to you. Basically, you need to stay 3 months (90 days) out of the Schengen area.
The only other option is to get a German long-stay visa, e.g. by marrying someone or finding a high-skill job (you need quite a lot of paperwork to get a permit) but that's about it. A somewhat more realistic options for other people (e.g. New Zealanders or Canadians) would be a working holiday visa but since the US does not participate in this programme, that won't help you.
Incidentally, you don't have a Schengen visa and couldn't get one as US citizen, what you are doing is a visa-free short stay.