A few examples that I find very intriguing:
1. type safe printf: https://github.com/mukeshtiwari/Idris/blob/master/Printf.idr
2. compile-time evidence that a runtime check will be performed: https://github.com/idris-lang/Idris-dev/blob/master/libs/bas...
3. Most of all, compile-time checking of state machine properties http://docs.idris-lang.org/en/latest/st/introduction.html
Think about being able to specify protocols in the type system and ensuring that your client and server meet the specifications. The example in that link is about user authentication. Imagine having a proof that the program can only get into a "LoggedIn" state by going through the authentication protocol.