By far the best examples of exciting animations in my opinion is Twitter, their iOS app is filled with them.
The moment you open the app you get the splash screen, most apps just hide the splash screen when it’s done loading. But twitter does something unexpectedly pleasant. It zooms into the logo and reveals the content behind it. They’ve designed it to be very smooth, and not take any longer than what it otherwise would be without the animation.
Another great example from Twitter, when you favorite a tweet, the star shoots sparks. It makes you want to favorite more tweets just to see the animation again. It’s a genius piece of UI animation which might actually increase conversion. Again, it’s very fast, it’s smooth and it doesn’t impede the user.
On the Nintendo website when you switch between views on the games page the button bounces, it makes the switch between the 2 views feel fast and smooth. The type of animation reflects its environment very well. It’s a gaming website, games are meant to be fun therefore they’ve decided to give the animation the feeling of something that's been pulled straight out of one of their games.
On the Apple website when you open the navigation on mobile, your screen is immediately taken over with fluid animations which relay context, and also make the interface feel fun and alive.
When you close the navigation the Apple logo and ‘bag’ icon pop back in as if they were hiding. The only down side to this is that it takes a long time for the animation to finish. From the moment you click the ‘hamburger’ icon to the moment you can click a button is about 2 seconds. For a navigation this is a long time.
This animation was added to a critical part of the user interface, which is a big no in the unwritten rules of web design. However a lot of websites with mobile navigation's are guilty of the exact same thing. This is, unfortunately, something we've gotten used to.