I've just noticed that Twitter allows you to export your tweets, under your account settings. They come in both human and machine-readable form. Not sure how long ago they added this, but it partially addresses one of my chief concerns about the service: that users stick many years of their lives into it without necessarily knowing that they'll ever be able to extract the information in future.

This is not easy to automate, though, so I'm still going to keep using Archive My Tweets for my own archive.