My USA passport pages are all full. On my entry into Germany I was reminded that entry to another country can be denied. My trip includes another country before returning home.
As of January 1st, 2016 your only option is to get a new passport. If you travel frequently you can request a 52 page passport book but if you run out of those you will need another passport again.
Assuming that "another country" is not in Schengen (if it is, there is no passport check between that country and Germany, and you will only have two stamps), you have very few options besides those already listed in other answers:
You can get a new passport quickly from the US Embassy when you're abroad (most likely this would be a temporary passport). They are quite fast with this, but it still might take 1-3 working days depending on the country, although I have heard from a guy who claimed to get a (temporary) passport the same day in the evening.
You can enter through a Schenghen country which is more lax than Germany, and then travel to Germany via intra-Schenghen flight or train. In Italy, for example, in half of my entries they didn't even stamp my passport on entry, and I assume they'd care less about having no space in your passport, while in Germany or Switzerland you can actually be refused entry.
If you have visas in your passport which you did not use, you might try to carefully peel them off. This is the last resort option for a life-or-death situation, as you have a very good chance to damage your passport completely, and might not be able to even return to the US. Note that some visas will have a stump "a visa was here" under it, and that removing a full-page visa is typically very difficult (but I've seen it being done, so it is not impossible).
Theoretically you may also be denied boarding by the airline if the agent at check in is curious enough, and gets through your passport and finds out there is no empty space there