Real-time desktop notifications for the Stack Exchange. Key features:

Real-time desktop notifications for the inbox of any user on the Stack Exchange.
Go to your inbox on click (this link can be configured).
Don't want to receive notifications any more? Expand the widget by clicking the icon in the add-on bar, and click Stop.
Unread notification count is displayed on the button.
Desktop notifications can be disabled.
Your StackExchange inbox is shown in the panel at the extension button.

Source code: GitHub Support: Stack Apps Chrome extension: Chrome Web Store