So, is there a facebook app where I can give something away to my fans. I want to run a contest where the person who recommends the most people, and the most people fan us win something cool. How do I do that?
I did something similar recently. My approach was based on fb:request-form. This way, facebook users could recommend something (doesn't matter if this is an app, a page or anything else) to their friends. Each of their friends receive a request, and you can track how many friends every user has brought. I can help you with the development, so pm me if you want to discuss more. If you are a developer, I can give you more details.
I have some collection of FB apps, Have a look and probably you may find the app you need there. Good day.