Mail.app used to have the ability to hide the unread count badge, as shown on this 37signals blog post:
It seems that Apple has removed this in OS X 10.8 Mountain Lion. Is there any way to restore this functionality and remove the badge?
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With the advent of Notification Center, the display of badges on any app can now be managed alongside display of banners, popups, and other notification-related settings.
Go to the
Notifications preference pane in
System Preferences, and select the Mail app from the list - the option to turn off the Dock badge should be there.
You may need to restart Mail for this change to take effect, though on 10.8.4 I do not.