The following is now a celebrated problem whose solution appeared in the general media. However I believe that some of the answers given are wrong.
Take a "large" disk with radius 1 and a "small" disk with radius $1/N$ where $N\in \mathbb N^*$. The small disk is rolling without gliding on the large disk and the question is: how many complete revolutions the small disk should do to go back to the initial configuration?
Let's see if I can give a satisfying geometrical answer, without invoking angular speed, frame changes, or hand-waving arguments.
Suppose the small circle has center $P$ when it starts, touching the large circle at $A$. After one full rotation of the small circle, its center will be at some point $P'$, while that point of the small circle which was touching the large circle at $A$ will be now at $A'$, with $A'P'$ parallel to $AP$.
But as one can see from the diagram below, after one full rotation the contact point is not at $A'$ but at some point $B$ and $x=\angle BP'A'=\angle AOB$. Only those points of the small circle forming arc $BA'$ (pink in the diagram) have been in contact with the large circle, hence the length of arc $BA'$ is the same as the length of arc $AB$ in the large circle, that is: $$ (2\pi-x){1\over N}=x, \quad\hbox{whence:}\quad x={2\pi\over N+1}. $$ After one rotation, then, the center of the small disk has covered a central angle $POP'=2\pi/(N+1)$: in order to complete a full $2\pi$ revolution around the large circle, $N+1$ rotations of the small disk are then needed.