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More than once I've installed Chatter Desktop on my Mac and found that when I start it the window is nowhere to be seen! What's interesting is that if you fire up Exposé you'll find that you can hover the mouse where the window should be and see a big blue rectangle; this rectangle is usually a highlight around the window and this led me to think that maybe the window was transparent.
I ran to the googles but couldn't find anything useful so instead decided to check out the application bundle and see if there might be something in there – it turns out my hunch was correct, and so here is the solution to a tedious problem. You need to edit a file in the bundle, typically it will be found in the directory /Applications/salesforce.com/Chatter Desktop.app/Contents/Resources/META-INF/AIR and the file is called application.xml. You can either navigate there via Terminal.app or you can navigate to /Applications/salesforce.com and right-click on Chatter Desktop and choose "Show Package Contents".
Once you've got the file open you need to find this line:
and simply change it to
Save the file, start up Chatter Desktop et voilà, thar she blows.
Note: You will need administrator access to be able to edit the file as it's not in your user path, either install Chatter Desktop to your home folder rather than /Applications or su to root in the terminal as I did and edit the file there (you'll need to setup a root password on your Mac to do this, if you don't understand what that means you probably shouldn't do it!).