Do you think there is a way to make a Facebook Like-button appear in the User menu of Drupal 7, right besides the My Profile and Logout links?
UPDATE: I'd like to add the button for my site and the User menu seems to be the best place.
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As an alternative to adding the like-button in the actual link items, maybe you should consider add it via a block or similar above/below your menu.
Checkout the Facebook social plugins integration module.