Solution
Notice that 720 = 6!, and 120 = 5!. Thus we can write this as:
1, 1, 2, (6!)!,((((5!)!)!)!)!.
Note further than 6 = 3!, so this can be written:
1, 1, 2, ((3!)!)!, ((((5!)!)!)!)!.
Finally, note that 1 = 1! = (1!)! = ((1!)!)! = … and likewise 2 = 2! = (2!)! = ((2!)!)! = … so we can rewrite the sequence as:
1!, 1!, (2!)!, ((3!)!)!, ((((5!)!)!)!)!.
This is the start of the Fibonacci Sequence (1,1,2,3,5,8,13,….) in which we apply the number of factorials (!) to each element equal to that number. So 1 has one factorial, 2 has 2 factorials, 3 has 3 factorials, and so forth. The next entry is thus 8 with 8 factorials:
(((((((8!)!)!)!)!)!)!)! = ((((((40320)!)!)!)!)!)!)!.