AA rarely comp their top tier (EXP). You will be very lucky if they comp you the Platinum (their middle level).
However, you can do their "challenge". Search AA forum and there are many discussion about either Gold or Platinum challenge. I used to think the best time to do it is at the beginning of the year. But someone told me if you do the challenge now, your status will last until the following syear. You need to verify this with AA though.