The problem with the backload method is that its still multiplying each number by 9. So even with backload, the x! is higher than 9^x after 21. x being the permutations that have already happened. If x is the same size in these two cases the factorial still wins (after 21). Since the starting number 9! is far more than 21, 9^x will never catch up to x! even if they used the same number of characters. Since backloading exponents actually uses 4 characters it falls even further behind.
so 9^(9^9) is less than 9!!! (we already proved that 9!! is more than 9^9 (same characters)). Since both 9!! and 9^9 are over 21, 9^(9^9) is less than (9^9)! Also 9!! is greater than 9^9, so we wouldnt use that anyways. 9!!! is greater than 9^9!! and still uses one less character.