User blog:ArtismScrub/how fast is this?

So, I've been devising a notation for nested factorials, and I have currently the following rules:

a![0] = a

a![b] = (a![b-1])!

(this means something like a![b]![c]![d]... = a![b+c+d+...])

a![b,1] = a![b]

a![1,c] = a![c,c-1]

a![b,c] = (a![b-1,c])![b,c-1]

It's easy to see that a![a] is comparable to f3(a), or roughly megafuga (a). However, I'm not so sure about the second entry. I'm afraid a![a,a] might not even reach f4(a). Here's what I found from evaluation:

n![2,2] = n![4]

n![3,3] = n![18]

n![4,4] = either n![134] or n![87]

n![5,5] = n![???]

Then there's also another issue: results seem to differ in the order they are evaluated. That's how I got 2 different results for n![4,4]. How should I state the order of evaluation in order to make this as powerful as it can be?

But however you order it, it only looks like a[a,a] only reaches f3(f2(a)) at best.

I was hoping for a linear array function that reached fω at 2 entries and fωω at its limits, but I would have accepted f4 at 2 entries and fω at its limits. After all, I would find a way to extend it... but it looks like this isn't getting anywhere.