Pali-n-drome this List
05AB1E, 4 bytes
Note that if only a single iteration was required (n=1), then the program would be the palindrome û€û.
Fû€û

Try it online
F       Do n times
 û      Palindromize the list
  €û    Palindromize each element in the list


If padding the input was still a required part of the program (11 bytes):
€R.B€RIFû€û

I couldn't find a shorter way to right-justify. Left-justification and centering were all easy, but this was longer for some reason. Using E or ² instead of I also works.