New notation idea?

Here is the basics:

{a} = a!!!...!!! with a nested factorial signs. Pretty basic for 1-entry arrays.

That's already a really powerful function. Just a single factorial sign is powerful by the layman's standards, but if we have a bunch of them, that's going to be amazing!

{0} = 0
{1} = 1! = 1
{2} = 2!! = 2! = 2
{3} = 3!!! = 6!! = 720! = 10^1,746.42
{4} = 4!!!! = 24!!! = 10^10^10^24
In general, {a} is about 10^^a

With one-entry arrrays, we already have created a strong function, but we're only in between f₂(n) and f₃(n) in the fast-growing hierarchy, so we need to keep going. Let's see what two-entry arrays look like:

{a, b} = {{{{...{{{a}}}...}}}, b-1} where there are {a, b-1} brackets. We've created a pretty strong recursion so far.
{a, 1} = {a}. Just like BEAF, if there are any 1's at the end, they can be removed.
{1, 2} = {{1}, 1} = 1
{2, 2} = {{2}, 1} = 2
{2, megafugafzgarboogagoogolplexian} = 2. If the first entry is 1 or 2, the whole array simplifies to just 1 or 2.
{3, 2} = {{{{3}}}, 1} = {{{{3}}}} = {{{10^1,746.42}}} = Template:10^^10^^10^^10^^(10^1,750)
{4, 2} = {{{{{4}}}}, 1} = {{{{{4}}}}} = {{{{10^10^10^24}}}} = 10^^10^^10^^10^^(10^10^10^24)
In general, {a, 2} is about 10^^^a

Not too bad - we've already achieved a pentation level function. But that's still only between f₃(n) and f₄(n) in the fast-growing hierarchy.

{3, 3} = {{{{3}}}, 2} = {10^^10^^10^^(10^1,750), 2} = {{{{...}}}, 1} = about 10^^^10^^^(10^1,750)
In general, {a, 3} is about 10^^^^a
In general, {a, b} = 10^^^...^^^a with b+1 arrows

Wow! With just two entries in our little arrays, we have created something that grows as fast as Knuth's arrows do. Thus, we have hit growth rate ω in the fast-growing hierarchy, starting from scratch! This is as powerful as the Ackermann function and three-entry BEAF. But the three-entry arrays grow faster quicker.

{a, b, c} = {{{{...{{{a}}}...}}}, b-1, c} with a bracket sets
{a, 1, c} = {a, a, c-1}

So {a, b, 2} has growth rate ω2, {a, b, 3} has growth rate ω3, etc. so the limit of 3-entry arrays is ω². That's the same growth rate as 4-entry arrays in BEAF! So far, we're one entry ahead of BEAF.

With 4+ entries, it's pretty much the same as normal BEAF (thanks for the suggestion Syst3ms):

signifies the rest of the array.

{a, 1, ..., 1, c+1, #} = {a, 1, ..., a, c, #}

So 4-entry arrays have growth rate ω³, 5-entry arrays have growth rate ω⁴, etc., and a-entry arrays have growth rate ω^a-1. So we can say that the limit of this is ω^ω. But beyond this, it's a little different.

{a | b} = {a, a, a, ..., a, a} with b a's
{a, 2 | b} = the following:
Stage 1 = {a | b} = {a, a, a, ..., a, a}
Stage 2 = {{a | b} | b} = {a, a, a, ..., a, a} with Stage 1 a's
Stage 3 = {{{a | b} | b} | b} = {a, a, a, ..., a, a} with Stage 2 a's
...
{a, 2 | b} = Stage a

Insane, right?? {a, 2 | b} has growth rate ω²^ω.

{a, 3 | b} = the following:
Stage 1 = {a, 2 | b}
Stage 2 = {{a, 2 | b}, 2 | b}
Stage 3 = {{{a, 2 | b}, 2 | b}, 2 | b}
...
{a, 3 | b} = Stage a

{a, 4 | b}, {a, 5 | b}, etc. should be obvious. Currently, the limit of this is ω^ω^2. But we need to add 3 entries on the left now!

{a, b, 2 | c} = the following:
Stage 1 = {a, b | c}
Stage 2 = {a, {a, b | c} | c}
Stage 3 = {a, {a, {a, b | c} | c} | c}
...
{a, b, 2 | c} = Stage b
{a, b, 3 | c} = the following:

Crazy, right?

Stage 1 = {a, b, 2 | c}
Stage 2 = {a, {a, b, 2 | c}, 2 | c}
Stage 3 = {a, {a, {a, b, 2 | c}, 2 | c}, 2 | c}
...
{a, b, 3 | c} = Stage b

That's insane, right?? The limit of this is ω^ω^3. That's not too bad!

We can generalize the pattern we have been making so far to have four, five, etc. entries on the left. In general, a entries on the left has a growth rate of ω^ω^a, so we can say that the maximum growth rate of this is ω^ω^ω, or ω^^3. Take that, ω^ω^3! Now, we will define:

{a | b, 2} = {a, a, a, ..., a, a | b} with b a's.
{a, 2 | b, 2} = the following:
Stage 1 = {a | b, 2}
Stage 2 = {{a | b, 2} | b, 2}
Stage 3 = {{{a | b, 2} | b, 2} | b, 2}
...
{a, 2 | b, 2} = Stage a (WHAT DO I PUT HERE? a or b)

{a, 3 | b, 2}, etc. should be obvious.

That's it so far, but I promise, I will add more soon!

Please, someone who is skilled with ordinals and FGH, TELL ME THE MAXIMUM GROWTH RATE of {a, b | c, 2}.