My favorite is bank accounts:

'Tell us your account number, as well the amount of a recent transaction on the account'.

Oh, so you gave me a personal check? Great, let me go deposit it, so that's a transaction that I know about. And I already have your account number (and routing information) from the check itself.

This isn't just to be able to drain all funds from the checking account (for which the numbers on the check alone are sufficient) but to reset the passwords on the account, which usually gives you access to any linked accounts as well.