User blog comment:Nayuta Ito/Multidimentional Ackermann Function (Part1)/@comment-5529393-20150301053909

The pattern goes

{0,n} ~ f_0(n)

{1,n} ~ f_1(n)

{1,0,n} ~ f_w(n)

{1(1)n} ~ f_w^w (n)

{1(1,0)n} ~ f_w^w^w (n)

{1(1(1)0)n} ~ f_w^w^w^w (n)

{1(1(1,0)0)n} ~ f_w^w^w^w^w (n)

{1(1(1(1)0)0)n} ~ f_w^w^w^w^w^w (n)

{1(1(1(1,0)0)0)n} ~ f_w^w^w^w^w^w^w (n)

so nested array notation goes up to epsilon_0.

The notion of nested array notation dates back at least to Chris Bird, although I wouldn't be surprised if it was thought of earlier, as it is a quite natural extension.