Depending on which Chick-Fil-a you want to go to, they actually give away a lot of free stuff. Which is great because they can be rather expensive at full price. Check out some of the events and free items at your locals restaurant. Remember these are exclusive to each store. Most of the stores keep to the same events, some name them differently and they put them on different days.
Market Street near Target
Oleander across from the mall
Mayfaire - no standard weekly promotions for Mayfaire, they always have something different going on.
Monkey Junction - i dont see any free ice cream but this store seems to cater more to the kids.