I used Excel to autogenerate all possible combinations of n1^n2^n3^n4, using n = 2..9, and found that I could form the following values which were within an order of magnitude of 1057:
I wanted more precision, though, so I added another exponent slot and repeated the process using the form n1^n2^n3^n4^n5.
To my chagrin, although I found many more combinations that yielded the above values, I didn't find any new values in between them. It would seem, then, that for any maximum value nmax, there is only a very limited number of values which can be formed as the result of n1^n2^..., where n is a whole number, and this holds regardless of how many n-values you use.
Is this something that's already well-known and has an obvious name? Like, "Oh, right, the Pulling-Dolbrach numbers are those numbers which can be formed as a result of successively exponentiating whole numbers."