What kind of entertainment do you have in mind?
Telling the boss that you are ready, willing and able to help other employees who could use the help and helping these employees is a perfectly acceptable kind of entertainment.
Asking the boss for a list of lower priority or long delayed projects that you could work on and that could put the firm on a sounder footing. and working on them is a perfectly acceptable kind of entertainment.
Going through the list of to-do's, identifying skills you might need to acquire to do these tasks and acquiring these skills is a perfectly acceptable kind of entertainment.
Keep in mind that the general rule that applies for whatever else you choose to do during your downtime, inactivity results in unemployment.
Why are you asking "Can the employee freely entertain himself, given the fact that he has completescompleted all assignments?" on this site? Why are you are not asking YOUR boss that question? If I were your boss, what do you think I would/should do to you if you were to askasked me that question?