User blog comment:Mh314159/new YIP notation/@comment-39585023-20190708013636

Here's a partial recursion of [1]. I haved learned that f1(x) is 3*2^(x-1) and therefore that f2(x) is stronger than tetration, I worked out that f3(3) is approximately 3^^^(2^^^4) or approximately 3^^^^3. So fx(x) grows faster than x↑xx. [1] turns out to be much bigger than I anticipated. It recurses the {0}0 function many times and {0}0(x) is fx(x). It looks to me like [1] reaches at least ω+1. I'm not sure what this implies for [2] or beyond.

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\[ [0] = 1 \] \[ [1] = \{ 1 \}_1 (1) \] \[ \{ 1 \}_0 (2) \] \[ \{ 0 \}_{f_2 (2)} (2) \] \[ \{ 0 \}_{f_2 (2) - 1}^{\{ 0 \}_{f_2 (2)} (1)} (2) \] \[ \{ 0 \}_{f_2 (2) - 1}^{\{ 0 \}_{f_2 (2) - 1} (1 + f_2 (2))} (2) \]\)