User blog comment:GamesFan2000/Hyper Tower Notation, Pt. 2/@comment-35470197-20190319012120

It is ill-defined because you do not have a rule to compute \(\#_2 \#_2\) and \(\# \#_2 \#_2\). For example, the rule 1 in not applicable to \(\#_2 \#_2\) because the rule allows only sequences of separators of length \(\geq 3\). The rule 3 is not applicable to \(\# \#_2 \#_2\) because the rule allows only sequences of length \(\geq 6\) whose fist (at least) three entries are \(#\)'s. As I pointed out in another blog post, your abbreviation is ambiguous compared to others. Why not writing in a recursive way? It is not so hard for you, if you can analyse notations beyond \(\omega^{\omega}\).

Also, you have not defined \(n\) in the rules 2 and 3.