Thanks everyone for the participation (and sorry for the delay in the answer).
There were actually $75 + 75 + 75 + 20 + 1 = 246$ of the little spheres inside the bottle. You can see them all lined up in the picture below:
Chris's idea of using the volume of the bottle as well as the volume a single sphere takes up is a good approach!
This problem is one example of what is called a "packing problem". These types of problems, even in two-dimensions can get fairly complicated fairly quickly! Perhaps we'll see another discussion about a "packing problem" in the future!