I am an American citizen visiting Berlin on business. Can I get a visa to China at the embassy?

You can try to get a China visa from Chinese embassy in Berlin, but the final decision is up to visa officer. It is hard to answer the question about China visa application in a third country because the visa rule varies in different countries. Some embassies/consulates asks non-resident applicants to provide residence permit or long-term visa, some embassies/consulates don't ask non-resident applicants to provide residence permit or long-term visa and issue China visa to them