Given a path to a directory, print the path to the biggest file in it
I find it weird that your function accepts a File but returns a String.  I would choose to return a File.

According to the JavaDoc for listFiles(),


  If this abstract pathname does not denote a directory, then this method returns null. Otherwise an array of File objects is returned, one for each file or directory in the directory.


Based on the documented behaviour, I would say that your if (current.isDirectory()) and if (files != null) checks are redundant, and I would eliminate the current.isDirectory() check.

It might be a good idea to do something about the possible SecurityException if you don't have permission to list a directory.

The throw new AssertionError() in the private constructor is an interesting idea that I haven't seen before.  Perhaps it would be better written as assert false;.