Electronic device with inner display and externally accessible input-output device

An electronic device may have an inner display mounted in a housing so that an image on the inner display is presented to an eye box through a lens for viewing by a user while the electronic device is being worn on a head of the user. The electronic device may have input-output devices that are operable on external surfaces of the electronic device. The input-output devices may be used to gather user input for controlling an external device while the electronic device is not being worn on the user's head. The input-output devices may include a touch screen display, buttons, and other input-output components. An input-output component may be formed on a movable member that can be moved between a stored position and a deployed position. A projector may project images onto nearby surfaces. A removable electronic device may be coupled to head-mounted support structures or other housing structures.