@WWMGT @mikeandallie It seems to be 4. The small-circle circumference gets spread out three times on the big circle, and after each time, it will have the same kissing point, but slightly rotated around the big circle. So 1+1/3 rotation for each time, making 4 altogether.
@WWMGT I thought it was going to be 6, which was being too clever by half.
@WWMGT An observer at B's centre, who turns to be always facing the point of contact, will see A turn three times, and will themselves turn once. An 'outside' observer will see the sum of both, i.e four turns.
@WWMGT Put an arrow from the center of the small circle to its initial point of contact. After one rotation of circle A, that arrow is pointing at an angle of 120 degrees (relative to the center of circle A). That is, it has revolved 360+120 degrees... etc.
@WWMGT Another way to visualise. First treat as gears: rotate big clockwise 1 turn so litte rotates 3 times anticlockwise. Now stuck litte to big and turn big back anticlockwise taking little round with it, it makes a 4th turn.
@WWMGT Three rotations of A as it travels around the circumference of B, 2πR/(2πR/3)= 3, plus the one rotation as the whole circle A travels all the way around B, so 4.
@WWMGT @mikeandallie geogebra.org/m/wjq2aatj After reading this quesiton I put together a little GeoGebra activity to show this in action. You can change the ratio of the radii. (I hope the rough explanations I included are not too terrible)