Looking at the picture of the £2 coins, imagine if the coin on top rotated one full time, but the bottom coin also rotated in the opposite direction - as if they were interlocking cogs, or a bit like the bottom coin is a treadmill on which the top coin turns.
In order for the top coin to rotate clockwise once, (i.e. 2 * pi * r), the bottom coin must also rotate anti-clockwise once. That is two full rotations. Now, if the bottom coin stays fixed (like a pavement and not a treadmill) as in the original statement of the problem, then is there some principle of physics that dictates the missing revolution of the bottom coin - because this bottom coin is now static - that says that revolution must come out somewhere? And that is why the top coin actually rotates twice? I'm thinking about a conservation law, but I do not know physics.