Extended extended chain notation

I will start with extended chain notation. For more extension, I will write numbers below the chain.

$a\rightarrow _{b}c=a{\xrightarrow[{b}]{}}c$

This looks like chemical equation, so I call {what is below the chain} {catalyst}. The catalyst makes number really bigger.

Catalyst must not be one. Some reactions require more than one catalysts.

$a{\xrightarrow[{1,0}]{}}b=a{\xrightarrow[{a{\xrightarrow[{a{\xrightarrow[{\cdots a{\xrightarrow[{}]{}}a}]{}}a}]{}}a}]{}}a$ (b stories)

Next, using the same rule, we can define up to $a{\xrightarrow[{1,x}]{}}b$.

Then,

$a{\xrightarrow[{2,0}]{}}b=a{\xrightarrow[{1,a{\xrightarrow[{1,a{\xrightarrow[{\cdots 1,a{\xrightarrow[{}]{}}a}]{}}a}]{}}a}]{}}a$ (b stories)

Keep going on until $a{\xrightarrow[{x,y}]{}}b$. At this stage, I think this is as big as $f_{\omega ^{4}}(x)$.

Next, three catalysts come.

$a{\xrightarrow[{1,0,0}]{}}b=a{\xrightarrow[{a{\xrightarrow[{a{\xrightarrow[{\cdots a{\xrightarrow[{}]{}}a,0}]{}}a,0}]{}}a,0}]{}}a$ (b stories)

See the pattern? As you know, definition can be up to n-entry.

This notation makes up to $f_{\omega^\omega}(x)$.

If you have a question, ask in a comment.

For comment, I will write generalized formula.

If the last number under the chain is not zero, it is same as extended chain (so you can ignore what is the left).

$a{\xrightarrow[{k,0}]{}}b=a{\xrightarrow[{k-1,a{\xrightarrow[{k-1,a{\xrightarrow[{\cdots k-1,a{\xrightarrow[{}]{}}a}]{}}a}]{}}a}]{}}a$ (b stories)