This question is pretty much 'what is a game engine'. Game engines are whatever software is needed to make a game, and there is no accepted checklist of what needs to be in such an engine for it to qualify.
That said, any simulation-style game will essentially do three things: acquire input, perform simulation, and present output.
For input you need to consider spatial peripherals such as mice, keyboards, joysticks, joypads, tablets, touch-screens, computer vision systems like Kinect, Wiimotes, etc. You might also consider audio peripherals such as microphones and indeed speech-to-text one layer up from that. The network is also another potential form of input.
For simulation, a lot of this is game-specific so it's hard to provide generic functionality that suits all games. But systems that may exist for this purpose could include physics simulation, pathfinding and planning algorithms, hooks for script-based decision-making, different types of time-based animation, and so on.
For output, the classic one is graphics - typically divided into the presentation of 3D models (which in turn tends to divide into environments and vehicles/characters) and 2D sprites and particles, but often with some overlap between the two and additional functionality for post-processing. There may also be support for video streaming support, multi-monitor displays, a GUI/HUD system, etc. Beyond graphics you have sound, usually including positional and non-positional sound effects, streamed music, and maybe filters to apply to the sound. Other forms of output include networking (again), force-feedback devices, log files (often more complex than you think), on-screen debugging aids and overlays.
There will also be some systems that exist outside of or parallel to the simulation which are useful for development - eg. tools for creating and manipulating game assets, persistence of game state (for save/load), script editors, internationalisation and localisation tools, test suites, etc.
How many game engines contain all of the above? Probably none of them. I expect some of the commercial offerings come pretty close however.
What would you have to add to Ogre to get a full game engine? Well, you find out what you need for the kind of game you want, see what Ogre provides for you, and add the rest.