I've actually come up with two different versions of the Joker's Time Lord past, and am tempted to include both of them, as well as a broken pocketwatch, and a tendency for whatever strange story he makes up on the spur of the moment (abusive dad, scarred wife, murdered sister, etc.) to be subsequently supported by evidence.

Dammit! I wanted to stop writing Doctor Who fic, but I love insane temporal convolutions! And this would be swimming in them!