OK, there's a Gmail account whose email I wish to have copy of in another Gmail account. Doing that with new messages is trivial, but there are about a thousand messages in there I want to backup too.
What doesn't work:
Settings/Accounts and Import/Import mail and contacts – you can't import from another Gmail account!
Settings/Accounts and Import/Check mail using POP3 – although you can set up reading another Gmail account via POP3, the Gmail server will not let you leave the read mail on the server. So you can use this to move mail to another account, but not copy it.
Settings/Filters then setting up a filter that catches everything, telling Gmail to forward it to the other account, and saying 'apply this filter now to all those existing messages' – it simply doesn't forward them although it will do other things to them all, like apply a label.
At the moment, I am pondering setting up a third account, then on the backup account, set up forwarding of all incoming mail to the third account, then use POP3 to get all the messages from the original account.
This will empty the original account (boo) but should result in there being two copies of everything: one on the backup, one on the third. Then on the original account use POP3 to read the third account.
Madness, but it might work…
Any other suggestions? Gmail's help doesn't seem to think that anyone might want to backup a Gmail account.