App Store keeps showing wrong badge count
Thanks, bmike, for the theory. It allowed me to find at least one practical solution.

Go into the App Store. Select Purchased → All. Now carefully scroll down the list. You will find one or more apps with a status of Install instead of Open. Go ahead and install them. If you have not upgraded to iOS 7 you may get a message that says that the latest version requires iOS 7, but you may download the latest compatible version. Touch Download and it will install.

These are the updates appearing in the badge count.