If you get this prompt every time you start iTunes, it’s probably an issue with the application contents vs the application’s code signature. Users seem to be encountering this sort of problem more frequently since Apple’s introduction of additional code signing, sandboxing, and GateKeeper functions in Mountain Lion.
Normally, this terminal command:
$ codesign -vvv /Applications/iTunes.app
should result in this:
/Applications/iTunes.app: valid on disk /Applications/iTunes.app: satisfies its Designated Requirement
If not, then the application package contents are probably mucked up. I recently encountered this situation as the command results showed a lot of extra files in the package. Probably leftovers from an update.
Most of the recommending fixes involve deleting iTunes and reinstalling from a fresh download. However, Mountain Lion won’t let you delete iTunes… say’s “can’t be modified or deleted because it’s required by Mac OS X.”
Some folk have had success by simply running the installer anyway. But in my case, the extra files weren’t removed. Instead I found I could right click on the /Applications/iTunes.app package, and “Show Package Contents“. Once inside the package, I could delete the contents. I simply deleted the entire “contents” directory, and then installed iTunes using a new download from the website. The terminal command “codesign” then generated the correct results and the Firewall prompts stopped repeating.
FYI. This occurred after a clean re-install of OS X 10.8.2 Mountain Lion on a 2012 MacBook Pro. I did a re-install to clean up problems from a number of app testing sessions and restore things to clean settings. After running available software updates from the App Store the iTunes / Firewall prompts began occurring each time I started iTunes. Most likely there was some issue with the App Store update for iTunes not cleaning up old files correctly.