Some countries require you to have six months remaining on your passport in order to visit. According to this list, Germany is not among them, but this note from the US State department says "U.S. citizens traveling on passports that expire in fewer than six months have increasingly been denied airline boarding or been detained upon arrival in certain foreign destinations, including popular European travel destinations in the Schengen area." This article echoes that.
The State Department's Schengen FAQ says "Some Schengen countries assume all travelers will stay the full three months allowed for visa-free visitors, meaning you may not be admitted unless your passport is valid for at least six months, regardless of the duration of your stay. This requirement may also apply if you are transiting a Schengen airport for several hours en route to a non-Schengen destination." Is Germany one of those countries?
Assume a US traveler planning to go to Germany for a few days with a passport that will be valid for more than three but less than six months after the return flight, with the departure flight too soon to go through the routine renewal process and be confident that a new passport would arrive prior to departure. Can the traveler wait until after the trip to do the renewal?
The Germany Page from the US State Department suggests the traveler can wait, but even that page is inconsistent about the number of blank pages required and the Embassy Messages link produces a 404 error, so the page is of questionable reliability.