「無限の無限乗」means ∞^∞ exactly. But its not aleph-one, the arithmetic of sets have two different types
rdinal arithmetic and cardinal arithmetic. The cardinal multiplication defined by Cartesian product. It likes this: RxR={<x,y>:x,y∈R},is a ordered pair, and the number of the “x” and “y” in this ordered pair is the dimension of the space contains all the <x,y>(so,RxR is 2d). The ordinal multiplication defined by ordinal addition:ω+1、ω+2......and you get ω+ω=ω•2(more accurately, ω•2 is the limit of {ω+n}. And you can get ω^2(the limit of ω•n), ω^ω(the limit of ω^n),but all of them are aleph-zero, because even ω^ω can be denoted by the union of all the ω^n (for every n=1,2,3......), and it is the union of countable numbers of countable sets that you can proof it’s also a countable set. But if you multiply ω numbers of ω by Cartesian product, you will get the ordered pair <n1,n2,......>(n1,n2,...∈ω,or positive integer) you can proof the set of all of this has the same cardinal with real numbers(R). And according to the context:[ルファスのステータスが一瞬で『∞』へと切り替わり、更にその∞の文字がどこまでもどこまでも表示され続ける。],the「どこまでもどこまでも」(endlessly) has a feeling of “endless process”, it’s more like the limit of ω^n(n from 1 to any finite ordinal number,increasingly) rather than the power set of aleph-zero[so,you can proof the aleph-zero’s power set has the same cardinal with all of the ordered pair <n1,n2......>(n1...is positive integer)]
So I agree with downgrade