Bows cannot be repaired using the Anvil, unless you have two bows.

Expected Result: Placing a bow in slot 1 of the anvil, and string (or sticks) in slot 2, should result in a partially repaired bow in slot 3.
Actual Result: Placing a bow in slot 1 of the anvil, and string (or sticks) in slot 2, results in an empty slot 3.
Solution: Adding the ability to use string (or sticks, your choice) to repair a bow.
Possible Drawbacks: None that I've found.