Awesome

This extension provides real-time desktop notifications for Stack Exchange's inbox.

Chrome extension

After installing the extension, visit Options and click on the "Grant Token" button. Upon click, a Stack Exchange authentication dialog will open. Grant the token, and the extension will autofill the user ID. Then, you will immediately get notified of the unread message count in your inbox.

The unread message count is also displayed on the extension button, and clicking on the extension button will directly show the contents of your inbox in a panel. There is a "mark as read" button to mark all messages as read.

(Alternatively, you can manually set the user ID, but then the extension will not be able to count the number of unread messages, and not be able to show the inbox content either.)

https://chrome.google.com/webstore/detail/desktop-notifications-for/ijglncoabcgieiokjmgdogpefdblmnle

Firefox add-on

The extension has been ported to a Firefox add-on since version 1.5!
Install it from https://addons.mozilla.org/en-US/firefox/addon/real-time-desktop-notificat/

Version 2.0 was rewritten using the WebExtensions API. For a data migration guide, see Firefox/README.md.

License

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.