You used to be able (when the EU was the EEC, and Schengen just an obscure town) to get a British Visitors' Passport at the Post Office, on presenting ID (not even photo ID). It required no proof of nationality, and was valid for travel to selected European countries for 1 year.

Doesn't exist any more.

Now, in the EU, you can travel to most other EU countries on an accepted form of national ID. This, of course, is only useful if you country has any form of national ID. Mine certainly doesn't...