The answer is $\frac{R}{r}+1$. The logic, as I read, is, the moving coin's center travels a distance of $2\pi{(R+r)}$ which is $\frac{R}{r}+1$ times the moving coin's circumference.
I get why the center moves $2\pi{(R+r)}$ units, but I don't understand how the argument works here, that is, why we consider the distance travelled by the center out of all things, and why we divide by the circumference of the moving coin to calculate the rotation. I can understand the argument holds but can not convince myself.
Distance travelled by the center of the radius $r$ coin is $d=2\pi (R+r)$, as you see.
That means the question is equivalent to "If a coin is rolled a distance of $d=2\pi (R+r)$ on a flat surface, how many rotations does it make?"
From here, we can note the circumference of the coin is $C_r=2\pi r$, meaning every $2\pi r$ units rolled on a flat surface is 1 full rotation.
So the number of rotations is the number of $C_r$ we can fit into $d$, $$ \frac{d}{C_r} = \frac{2\pi (R+r)}{2 \pi r} = \frac{R+r}{r} = \frac{R}{r} + 1 $$