In Illustrator, a simple Pattern Brush will work. Merely drag the art to the Brush Panel and choose Pattern Brush when asked, then just click OKAY. Draw a circle, and click the brush you just made.
If you want a bit of spacing between the iterations, extend the no-fill, no-stroke, rectangle in the brush artwork and then update the brush in the Brush Panel by holding the Option/Alt key down as you drag new artwork on top of the existing brush in the Brush Panel.
The only way to control the amount of iterations is to adjust either the size of the circle, or the size of the brush.
If you double-click the brush artwork in the brush Panel you can adjust the scale of the brush. Reducing the scale of the brush will allow for more iterations.
