User blog comment:Ikosarakt1/Hardy hierarchy up to psi(K)/@comment-25418284-20140617140300/@comment-5150073-20140617182459

It was actually discussed here. Your variant of the rule won't work because we can take $$\beta = \omega$$ and $$\alpha = \omega^2$$. By that rule $$\omega+\omega^2 \neq \omega^2$$.

The really proper variant of this rule, in terms of transfinite ordinals, in Cantor's normal form rule. But it is long and complicated to write, making unnecessary complexity to the notation!

I don't think that $$(\alpha+\beta)[n] = \alpha+(\beta[n])$$ without restrictions makes the notation ill-defined, it just allows minor offsets which doesn't change the growth rate. Hyper-E notation is well-defined, but it allowed things like #*#^#, and # is isomorphic to $$\omega$$, so I don't see the problem.