I had come to the conclusion that if one lens has twice the focal length of another, that means it makes everything look twice as big. Conversely, if a lens has half the focal length, you see twice as much stuff.

But is that exactly true? Or is it only an approximation? In particular, does this still hold for extreme wide-angle lenses? Will a 10mm lens show me twice as much stuff as a 20mm lens? Would a 1mm rectilinear lens (assuming such a thing even existed) show me 10x more view again? Or is this relationship only approximately valid for long focal lengths?