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 after you used up your 90 days 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.
Finally, you do not need to go back to the US or any particular country (say your country of residence), you just need to stay out of the area. Three months in Croatia, the UK, or Turkey would be enough to get another 90 days of visa-free stay.
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.