Here, a!! is defined as (a!)!.
123!!!!!! is absolutely gigantic. I think you’d need more particles than there are in the universe if you were to write it down in ink.
You can’t therefore compare the numbers directly. I conject that there is not a number class that can do this.
What you can do, is to consider the quotient 123!!!!!! / 456!!!. Many of the multiples will be similar, so you can cancel them. Note also that trailing ! will cancel. This is because x > y implies, and is implied by x! > y! where x and y are positive integers.
Eventually you’ll reach a point where you can evaluate this as being less or greater than 1, so yielding your answer.
I can tell you on inspection that 123!!!!!! is larger since 123!!! is larger than 456.