Owing the Joker
When you’re inside the walls of Arkham Asylum, Joker says you need to be free so that you can “continue your collision course with Lady Arkham.” He’ll tell you that the best way for you to escape is to use the phone and to make a phone call. A phone call to the right person is all that you need to get out of this place. Of course, in return for this information, he wants you to owe him a favor after being released from Arkham.
If you agree to this, Joker will remember this and smile a bit maniacally. Nothing is really revealed as to what the favor may be, but our guess is that this will all come to light in the final episode.
Telling Joker that he’s never going to get out of the Asylum and rejecting him of his favor sees him also remember your decision, but doesn’t lead to a major change at this moment in time.