OSX Users Only (unsure how to fix this using the PC - sorry in advance to all of those Windows people)
Further, this will not permanently solve the issue but is a workaround to get rid of the pesky "1" (or however many numbers appear next to the Applications menu item) without removing all of your applications and re-syncing (which does not always work as you will come to understand)... Warning: You will need to have some computer understanding to rid the pesky update notification!
Lets start with the problem shall we (there seems to be MANY posts with the same issue out there and hopefully I can shed some light on the situation). Please share my finding with anyone having this problem...
We all know that the App store is having their share of versioning issues - (i.e. the developer has a more recent application in queue than what is being displayed on the App store). Here is where the problem resides... iTunes is able to see this update, however it is not "ready" to be pushed to your computer (this is also true for the App Store on the iPhone/iPod Touch). Thus fooling it to think there really is an update - when in fact, its not ready for download.
Simple fix right...? Wrong. This issue has been around since the App store's inception and there is no provision in iTunes (not yet anyways) to ignore the soon-to-be-released-update. The update mechanism writes a key to the com.apple.iTunes.plist which you can clearly see if you follow this link:
http://home.carolina.rr.com/elachowitz/plistscreenshot.html
Notice the obvious: ipau stands for iPod Application Updates and there is that pesky "1" we so very much want to rid ourselves of...
An easy fix would be to delete the entire plist file from your preferences directory (this is the path: /Users/<USER NAME>/library/preferences/com.apple.iTunes.plist). However, I recommend that you install an application like PlistEdit Pro (
http://www.fatcatsoftware.com/plisteditpro/) to view the contents of the file and deleting the key ONLY (removing the whole file will cause iTunes to recreate a new one, thus causing you to have to re-enty some settings). Further, I'm sure there are some free Plist editors out there - you can do some diligence.
Until this gets fixed by Apple, the "badge of death" as I have seen it referred to, will not go away anytime soon...
Hope this helps.
-Eric