Chest Freezer TOO full?
Your friend is correct. The only way your chest freezer is too full is if the lid does not close properly. I imagine you could construct a way to fill the freezer such that some sectors of the freezer may not reach the correct temperature, but that would be a real challenge on most modern chest freezers.

There are even some advantages to keeping your freezer full (see this question):


  
  if you need to survive intermittent power outages, having a lot of ice or food in the fridge will keep it cool longer
  a larger amount of food in the fridge/freezer will make it easier to chill or freeze new food more rapidly, which may sometimes aid in
  food safety/preservation
  similarly, having more food may help to minimize small temperature fluctuations when opening the door frequently, again perhaps aiding in
  food safety/quality in some cases
  


There are some disadvantages to having a full freezer, like it may take longer to find what you are looking for or if you are like me it may be hard to even know what is in your freezer. That can lead to losing track and having some food stuffs that could be years old stuck at the bottom of the freezer. But in general, there's nothing wrong with a full freezer.