If you're writing a formal letter, and need the other party to reply as quickly as possible, how can you do it in the most polite, eloquent fashion? No matter how I try to do it, it always sounds too demanding.
Although I understand that you are very busy, I would appreciate it if you could provide a response as soon as possible. I very much appreciate your time.
My exact wording might change according to the circumstances – How well do I know the person? How important is it that I get timely feedback? Is this the first time I'm asking, or the third? What is the situation? Am I planning a reception, or arranging for payments on a loan that is about to go into default?
Those considerations aside, the generic wording I've included at the top of my answer is about as polite as I can muster.