Chrome's notification system works best when combined with extensions and Web Store installable or hosted web apps. When a user installs an extension or web app, they are asked once whether they want to give the app certain permissions, such as local storage, location, or desktop notifications. Once they give that permission, it's permanent until the app is uninstalled again.
So you should consider that when you design for Chrome notifications. The best use case is when you're launching an app in the Web Store, as you'll have a 100% guarantee that your audience will be able to benefit from the feature.
One of the most valuable reasons to have Chrome notifications is to add value as an application. You can use background processing to pay attention to something in the background even if the user doesn't currently have your app loaded, and then pop up a notification to alert the user. If your app has behaviour that takes a while to execute, that's a great way to tell the user some task has completed. Or you can let them know when they receive a new message, for example.
In fact, Google recommends using these features to offer a "rich experience". From Google's "Thinking in Web Apps" design principles for Web Store applications:
A web app offers a rich, engaging experience for the
user that is indistinguishable from native desktop applications.
typically should meet or exceed users' expectations of how functional
a web app can be. Web apps should use and exploit features, visual
clues, and interaction paradigms previously found only in desktop
For example, using modern techniques and browsers, web
apps can offer desktop notifications, geolocation, beautiful
typography, drag-and-drop, high quality graphics, and more. They can
use cross-origin resource sharing to consume content from remote
sites, and they can use WebSocket to show live content and support
real-time communication across the web. Ideally, the user forgets
about the browser and focuses solely on the application.