First few things that come to mind:

  - Access to Camera & Microphone
  - Access to USB devices
  - Push Notifications
  - Geolocation
  - Bluetooth

"deep" is probably the subjective term here.

Printer dialogs in browser [_we hates it_].

Access to raw USB devices is a good example I suppose.

Oh sweet. What could possibly go wrong?

Not much, really. Unless the device has been created to work with WebUSB, you first need to manually add a udev rule (on Linux) or replace the driver with WinUSB (on Windows) before you can access it through a browser. Even then, the user needs to select the device from a list, like you would when giving a site access to your camera.

Right, why would we want applications to be able to use hardware anyway?

Does Chrome OS qualify?