User blog comment:Eners49/New Notation Idea?/@comment-25601061-20180619205145/@comment-30869823-20180623131101

Isn't the growth rate ω+3 ? only the brackets with 2 elements iterate.

Proof by induction: \( [a_1,a_2,\cdots,a_n]\approx f_{\omega+2}^{2^{n-2}}(a_n) \) for n≥2

n=2 : \( [a,b]\approx f_{\omega+1}^b(a)\appox f_{\omega+2}(b) \)

n+1 : \( [a_1,a_2,\cdots,a_{n+1}] = [[a_1,a_2,\cdots,a_{n-1},a_{n+1}],[a_1,a_2,\cdots,a_{n-2},a_n,a_{n+1}],…]\approx [f_{\omega+2}^{2^{n-2}}(a_{n+1}),f_{\omega+2}^{2^{n-2}}(a_{n+1}),\cdots,f_{\omega+2}^{2^{n-2}}(a_{n+1})]\approx f_{\omega+2}^{2^{n-2}}(f_{\omega+2}^{2^{n-2}}(a_{n+1}))=f_{\omega+2}^{2^{n+1-2}}(a_{n+1})\)