My daughter asked me this question, after I pointed out that some Julia sets are a Cantor dust.
For a point not in the Mandelbrot set, the corresponding Julia set is a Cantor dust with an uncountably infinite number of disconnected points. There are also an infinite number of baby Mandelbrot's, each with an infinite number of bulbs.
Can you prove that the set of bulb's is countably infinite?
I think the answer is that every bulb can be associated with an external ray, and it seems that the known landing rays are all associated with rational numbers; the set of rational numbers is countably infinite. Perhaps if a landing ray is a real number, it would be associated with an infinitely small baby Mandelbrot, which is no longer a baby Mandelbrot because it has zero area? But another possible answer may also be because bulbs have area>0, and perhaps you can't have an uncountably infinite number of things with positive area.