User blog comment:DrCeasium/new hyperfactorial array notation/@comment-5529393-20130416153852/@comment-5529393-20130418182133

It looks like n![1,1,1;1,3] is where we start to disagree. As you say, nesting the third entry takes you to the next epsilon number. So

n![1,1,1;1,2] ~ f_zeta_0(n)

n![1,1,1;2,2] ~f_epsilon_(zeta_0+1) (n)

n![1,1,1;3,2] ~ f_epsilon_(zeta_0+2) (n)

and so on. So n![1,1,1;1,3], which nests over the fourth argument, will be about f_zeta_1(n), not f_eta_0(n).