Probably one of the most popular uses of a device like the Chromebox would be as a Home Theater PC, streaming music, movies, and other media to an HDTV. Intel's integrated HD Graphics engine has come a long way and clean output to an HDTV is a no-fuss no-muss sort of setup. Just plug it in and at 1080p (or 720p if that's your TV's limit) everything just scales correctly and is cleanly centered on the screen--4K too for that matter, via DisplayPort.
So firing up a Chromebox for Netflix, Amazon Prime Instant, Pandora, or what have you, is an excellent way to experience the latest in media services, including 4K media if you can find it.
X-Men: Days of Future Past 4K Ultra-HD Video Trailer - No sweat for the ASUS Chromebox.
Here we're streaming a movie trailer, X-Men: Days of Future Past at 4K resolution. (Granted, the viewing area is scaled down to fit the 1080p desktop for this screen capture.) In full-view mode, the experience is quite good, and the Chromebox didn't break a sweat.
Chrome Calculator, Google Books, and Google Keep
Here you're a looking at the calculator, Google Books, and the Google Keep app, all of which will run in offline mode, provided (in the case of Books) you've already downloaded content. Google's Keep notes app will give you access to any offline files that you might have or can add locally, and of course the Calculator app just works.
Pixlr Image Editing App
However, Pixlr, a really nice photo editing app offers online only functionality. It's a pretty decent alternative to tools like Photoshop, but you need to have an internet connection because the product is completely web-driven.
The ad-supported Mortal Kombat 3D is also a game that you need a full internet connection for. (It was actually kind of fun going back with this arcade classic, by the way). So, while there is a fair degree of offline functionality with the Chromebox, online connectivity is important to capitalize on the full experience.