Disclaimer! This is not definitive comparison with all the available tools. The comparison is done with the main popular tools. The opinions can be subjective. The whole idea of the section is to help users to choose the tools which best suits to their needs.
The Chrome extension supports Github and Github enterprise server. However, you can enable only one Github Enterprise server or github.com account. As is shown in the screenshot the extension shows the repository name, Pull request/Issue name and why the notifications is shown. After clicking on the extension you will go to the Pull request/Issue.
Users can enable RSS feeds from github.com or Github Enterprise server by clicking to "Subscribe to your news feed" as shown below. Users need to know that RSS will push exact same things that users will see in their Github or Github Enterprise home page. This notifications will not include Issue/Pull Request events like opened, closed, merged (only for Pull requests) and comments.
Then you can install one of the RSS feed notifier browser extension or a desktop application and get the notification popups
Octobox is impressive tool for github.com accounts. It does not show notifications popups but it keeps the history of all notifications in github in well organized and user friendly way. One can say that github-notifier and Octobox can be used together, one of real time notifications and the other for the history.