You may need to file an application in the Copyright Office as a prerequisite to filing a lawsuit for copyright infringement in the US. It is not required in order to send a C+D or a DMCA takedown notice, but failure to file an application is often an initial defense people use to throw you off their scent.

I think if you published a game on iPhone App store, then you can file a copyright application to protect the audio-visual elements and the underlying code (technically both are protected under 1 combined work). Remember you have to disclaim licensed works that are included in the App (e.g. open source middleware), but I would not bother registering the GDD b/c it will likely anyone can look at the iPhone game and extract the basic ideas and game play mechanics without looking at the GDD. So it would be kind of pointless b/c it would only prevent a direct rip-off (same art, same story, characters etc.).

Good luck!

@Tom Sloper
Apparently there is quite clear answer to my question (depends on a country, but generally no notice is also giving me copyrights). Thanks for pointing it ;)