The UK does. I've never been asked to prove / provide evidence for anything in order to be able to vote.
Letters get sent out to every address in that council's area with options for freepost, free phone call or online registration, and each letter has a unique registration number. I think you have to respond in order to qualify to vote in the next election, whether you were previously registered or not.
The process gets integrated into council tax registration as well so if you move house within the community, the council tax guys ask if you want to be registered to vote for the next election (yes/no question, that's it).