Two of the iPhone's default apps, Weather and Stocks, create little widgets in the Notification Center. They aren't exactly notifications, can display graphics (weather) or animations (stock ticker), and don't disappear after being tapped.
I have not seen any other apps that do this. Do any other apps use widgets like this? Or is this exclusive to Apple's apps?
3rd party widget development is not currently supported by the standard iOS SDK (as of iOS 6). You can not find any other apps to show a widget similar in use to weather and stocks in the notification center.
You could jailbreak the device to run custom widgets, however. The jailbreak community has created many widgets.
This is exclusive to Apple's apps. No other app is allowed to do that. Another widget that does this (also Apple's), is the Share widget which allows tweeting and posting to Facebook (not pictured here) right from Notification Center:
