Sometime last year, Facebook began showing the combined number of friend requests, inbox messages, and notifications within the browser tab. The figure on the tab updates whenever new alerts are received to keep the user up to date on any changes even if the Facebook tab is not open. But this does not include chat messages which usually updates more often than notifications.
There is an Opera extension that fixes this. The extension shows the number of new chat messages you have in your Facebook chat window on Facebook’s favicon, which is arguable a better place for displaying notification than the tab. Tabs resize when there are large number of tabs open and that truncates the title which can hide the notification figure. The favicon is always displayed, even when the tab is pinned.
I’m aware there aren’t many Opera users here, so I tried to find similar extensions for Firefox and Chrome. Unfortunately, there are none at the moment.
There is a Tab Badge add-on for Firefox that basically shows the same message and notification count that Facebook does by default, but on the beginning of the tab title. This ensures that the count number is not hidden when the tab resizes. This also works on pinned tabs. But chat message count is not available.
Tab Badge works on other websites too such as Twitter, Gmail, Google Reader so it not entirely useless. Gmail, of course, has the notification-on-favicon functionality as a Labs feature that you need to enable.
For those who don’t mind running a dedicated desktop program for receiving Facebook notifications, there is FBNotifier.