But, my short answer would be to leverage your existing product/service to convert your active community into FB fans. Those will most likely be your most engaging FB fan audience and easiest to become fans with the right incentive. From there, your networks of fans will only start to grow.
Sorry I couldn't give you a specific answer, but without knowing your business I couldn't provide a suggestion. In my own community, we encourage the players of our games to become facebook fans through virtual currency, bragging rights, and other incentives.
Is it to retarget visitors to your site through Facebook ads? In that case, a Like button on your site would work well.
Is it because your app has a viral element that requires appearance on profiles/News Feed to grow? Then you should be incentivizing users to friend you by offering them something of high perceived value in return.
Or do you simply want a high number of fans as a growth metric to show off? For that, there are numerous enterprises online that will sell you as many fans as you want at a reasonable price.
