User blog:Googleaarex/Aarex's Ultra Array Notation

[1[1][2]2] = [1[1[...[1[1]2]...]2]2] = [1[1[...[2]...]2]2] has level BHO

[1[1[1][2]2]2[1][2]2] has level theta(e(W+1),1)

[1[2[1][2]2]2[1][2]2] has level theta(e(W+1)+1)

[1[1[1[1][2]2]2[1][2]2]2[1][2]2] has level theta(e(W+1)*2)

[1[1][2]3] = [1[1[...[1[1[1][2]2]2[1][2]2]...]2[1][2]2]2[1][2]2] has level theta(e(W+2))

[1[1][2]1[2]2] = [1[1][2][1[1][2][...]]] has level theta(e(W*2))

[1[1][2]1[3]2] has level theta(e(W^2))

[1[1][2]1[1[1][2]2]2] has level theta(e(e(W+1)))

[1[2][2]2] = [1[1][2]1[1[1][2]1[...[1[1][2]1[1[1][2]1]2]...]2]2] has level theta(z(W+1))

[1[[1[2]2][2]2][2]2] has level theta(phi(w,W+1))

[1[1[2]2][2]2] has level theta(phi(W,1))

[1[1[1][2]2][2]2] has level theta(W_2)

[1[1][3]2] = [1[1[...[1[1][2]2]...][2]2][2]2] has level theta(e(W_2+1))

[1[1][[1[2]2] ] |undefined2] has level theta(W_w)

[1[1][1[2]2]2] has level theta(W_W)

[1[1][1[1] [2]2] 2] has level theta(W_W_2)

[1[1][1] [2] 2] = [1[1][1[1] [...[1[1] [1]2]...] 2] 2] has level psi(psi_I(0))

[1[1][2] [2] 2] has level psi(e(I+1))

[1[1][[1[2]2] ] [2]|undefined 2] has level psi(W_{I+w})

[1[1][1] [3] 2] has level psi(psi_{I_2}(0))

[1[1][1] [[1[2]2] ] |undefined 2] has level psi(psi_{I_w}(0))

[1[1][1] [1[1] [1] [2] 2] 2] has level psi(psi_{I_I}(0))

[1[1][1] [1] [2] 2] has level psi(psi_{I(1,0)}(0))

Limit is [1[1][1] ... [1] [2]  2] or psi(psi_{I(w,0)}(0))