Cayley Mathematical Olympiad 2006 Q6

A mathematician has a full one-litre bottle of concentrated orange squash, a large container and a tap. He first pours half of the bottle of orange squash into the container. Then he fills the bottle from the tap, shakes well, and pours half of the resulting mixture into the container. He then repeats this step over and over again: filling the bottle from the tap each time, shaking the mixture well, and then pouring half of the contents into the container.

Suppose that on the final occasion he fills the bottle from the tap and empties it completely into the container. How many times has he filled the bottle from the tap if the final mixture consists of 10% orange squash concentrate?