User blog:Eners49/New Notation Idea? (Attempt 2)

Here is a new notation I am thinking about. Someday I hope to make it go beyond w^w in the fast-growing hierarchy. When I saw PlantStar's Notation and how easy it was to create your own notation, I got the inspiration to create my own. However, the first time I tried, I utterly and completely fucking failed. I extended upon Graham's function, which has growth w+1, but I did a bunch of complicated stuff and only got to w+9 :( For my second attempt, I will try to make my own version of BEAF, based off of the factorial function. Right now, I have not developed very much and my notation is not very strong yet, but it will be! I promise.

Here is the basics: With one-entry arrrays, we already have created a strong function, but we're only in between \(f_2(n)\) and \(f_3(n)\) in the fast-growing hierarchy, so we need to keep going. Let's see what two-entry arrays look like: Wow! With just two entries, we have created something that grows as fast as Knuth's arrows do. Thus, we have hit growth rate w 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 in my notation are way crazier. That's it for now, but I promise, I will extend this further and further.
 * {a} = a!!!...!!! with a factorial signs. Pretty basic for 1-entry arrays. Similar to Aarex's warp notation.
 * {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
 * {a, b} = {{{{...{{{a}}}...}}}, b-1} where there are a 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} = {{}, 1} = {{{{3}}}} = =  = about 10^^10^^10^^10^^(10^1,750)
 * {4, 2} = {{}, 1} = {{{{{4}}}}} = = 10^^10^^10^^10^^(10^10^10^24)
 * In general, {a, 2} is about 10^^^a
 * {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
 * {a, b, c} = {{{{...{{{a}}}...}}}, b-1, c} with a bracket sets
 * {a, 1, c} = {a, a, c-1}
 * With this, we're somewhere in between 3-entry and 4-entry BEAF.

Please, anyone who has experience with fast-growing hierarchies and stuff, tell me the growth rate of my 2-entry and 3-entry arrays. I suck at fast-growing hierarchies and probably everything I have written is complete and utter nonsense, and my 2-entry arrays probably just as weak as my 1-entry arrays. Someone pls help me