User blog comment:Mh314159/FOX notation/@comment-39585023-20191205111828/@comment-35470197-20191205120801

The limit roughly corresponds to (ω^ω)×2. Repeating this process, you can achieve (ω^ω)×n at the n-th repetition. In other words, your construction from f(x) to f(x) effects to ordinals as +ω^ω.

Then you will understand that diagonalising the n-th repetition, which corresponds to (ω^ω)×n, you will achieve (ω^ω)×ω = ω^{ω+1}. In this way, we create a stronger recursion. But of course, this construction will be stuck at a stronger level such as ω^{ω×2}.