User blog comment:Simplicityaboveall/The Construction of Extremely Large Numbers/@comment-11227630-20160726150520

Your idea is similar to BEAF. Also, your notation has similar problems to BEAF.

The x -> [1@x/10] function (where x is a []-expression) "map SGH into FGH", which is similar to the & operator in BEAF, and that makes your notation as strong as legion arrays in BEAF. I think that's your aim of this notation.

However, how to map numbers into ordinals correctly is a main problem, and you don't explain it in detail. For example, $$R_{10}^\omega(10\uparrow\uparrow11)$$ could be $$\omega^{\varepsilon_0}=\varepsilon_0$$ since $$R_{10}^\omega(10\uparrow\uparrow10)=\varepsilon_0$$; it could also be $$\omega\uparrow\uparrow11$$; but I think neither is what you want.

To solve the problem, we need some rules to determine $$R_{10}^\omega(n)$$. (And the same as BEAF. In BEAF we need some rules to determine the & operator, or higher arrays than tetrational arrays)