I'm currently in the process of filling in the ESTA form for a holiday in America. The section "Country of Birth" is required; I was born in Germany, but to British parents on a British RAF base, and I'm British and have a British birth certificate. I have no legal connection to Germany at all. Do I put my country of birth as Germany or UK?
You were born in Germany.
The UK does not have sovereignty over its military bases in Germany. They are still part of the host country, but under extraterritoriality, similar to embassies. For most purposes Germany does not have jurisdiction there, but still asserts sovereignty.
There are separate questions on any former or present nationalities.