How many times do human hands touch milk between the time it leaves the cow until it reaches your table?

Using modern milking techniques and equipment, human hands should not come into contact with the milk. Milk is transferred through pipes from the cow (milking cluster) to processing (pasteurisation / sterilisation) to the packaging material.

Even before modern milking techniques and equipment hands rarely touched the milk. When milking a cow by hand, you don't get your hands covered in milk -- and there's no reason to touch it while pouring it from the milk bucket to a larger container.