Drawing sprites/graphics on screen without the use of a window.

I know this is a long shot, but would it be possible to create a game that doesn't use a system window? Or would I just have to learn how to do this in OpenGL or something?

I wanted to program something like a desktop companion. Would it all be possible to make a window transparent, or make the the boarder invisible?

OpenGL and other rendering apis render to a window. You can however create borderless windows and set transparency as well. I don't believe this is anything you can use Unity for though :/