User blog comment:DrCeasium/Hyperfactorial Notation - Take 2/@comment-5150073-20130330145048

Look at these comparisons:

n!^1_1 < n^n

n!^2_1 < n^^n

n!^m_1 < {n,n,m}

n!^2_2 < {n,n,1,2}

n^3_2 < {n,n,2,2}

n^m_2 < {n,n,m-1,2}

n^2_3 < {n,n,1,3}

This is because the rule 3 simulates 3rd entry in Bowers' notation (plugs back the polyponent, n times), and rule 2 plugs back the 3rd entry. So your notation is approximately as powerful as the previous one.