Some examples: A voice recorder. A calculator. A camera application. A compass. A drawing program. A photo editor. A program that auto-tunes your voice. Games.
As a matter of fact, even for those non-app websites that offer an app - I find the app usually offers a better experience compared to the site, but like you - I'm not installing a separate app for each one of them.
The experience might be better, but there might be other valid reasons not to use them. That's a trade off vendors have to have in mind as well. Native apps are not always the optimal solution given certain constraints.
In my ideal world, I'd prefer a native app for all things. The web sucks in my opinion. Honestly, I wish there were a browser that dealt exclusively in native GUI elements and other computing resources. I'd also want some very finely-grained native resource permission controls.
I don't every want to layout my UI by saying "div div div div div div div". I want to say "window menubar toolbar grid treeview datagrid statusbar" and then I want it to show me the native representation of those things.