User blog comment:Ubersketch/I need some help with this unnamed notation./@comment-5529393-20180919004507

So {a+[1]b} = a+b.

It's not clear to me what you mean by "{a+[1]b[1]c} = {a+[1]b} c times (i.e. {a+[1]b[1]c} uses {a+[1]b} as the input for a c times)", but my best guess is that you mean take f(x) = {x+[1]b} and apply f c times starting from a. This gives {a+[1]b[1]c} = ac+b.

Then it is not clear what {a+[1]b[1]c[1]d} is, but I will assume you want to take f(x) = {x+[1]b[1]c} and apply f d times to a. This gives you ac^d + C where C is constant with respect to a.

Continuing in this fashion, adding more letters give ac^(de) + C, ac^(def) + C, and so on.

I assume you want {a+[2]b} = {a+[1]a[1]a...[1]a} with b [1]'s, not a [1]'s. (Otherwise b is not doing anything.)   Then for b > 2 we have {a+[2]b} = a*a^(a^(b-2)) + C.

After that I don't know what you want to do; do you want the next step to be {a+[2]b[1]c} or {a+[2]b[2]c}, what exactly does that equal, etc.