User blog:GamesFan2000/Extended E-star(E*) Notation

A day or two ago, TechKon, formerly SpongeTechX, defined what he now calls E-star notation. The definition came with a small contribution from me. Originally, E*(0) was undefined, but I found that making it equal to 1 made the definition easier to understand. As a bit of a bonus, it made the function slightly stronger in the context that the results were larger. The true strength of the function didn't really change, but Tech thought it was a good idea and went along with it. The definition of the function is as follows:

E*(0)=1

E*(n+1)=E*(n)+E*(n)E*(n)

The following extension was made by myself, completely independently of TechKon. I don't mean to infringe on him, I just want to provide a proof of concept of how an extension of this might work.

Definition
E*(0)=1

E*(n+1)=E*(n)+E*(n)E*(n)

E*(n, 0)=E*(n)

E*(n, m+1)=E*(E*(n, m), m)+E*(E*(n, m) m)E*(E*(n, m), m)

E*(n, m, ...0)=E*(n, m, ...)

E*(n, m, ...x+1)=E*(E*(n, m, ...x), E*(n, m, ...x), ...x)+E*(E*(n, m, ...x), E*(n, m, ...x), ...x)E*(E*(n, m, ...x), E*(n, m, ...x), ...x)

Examples
E*(2, 2)=E*(E*(2, 1), 1)+E*(E*(2, 1), 1)E*(E*(2, 1), 1)

E*(4, 4, 5)=E*(E*(4, 4, 4), E*(4, 4, 4), 4)+E*(E*(4, 4, 4), E*(4, 4, 4), 4)E*(E*(4, 4, 4), E*(4, 4, 4), 4)

Explained in Words
We extend ordinary E* notation by allowing arrays of numbers to be valid within the parentheses. For an array ending in 0, crop the 0. If that doesn't apply, then reduce the final entry by 1, and replace all previous entries with this new array, remembering to use the E* function on those arrays as well. You will then perform a hyperoperation; the hyper-n level is equal to the expression you've built, and the number the operation will be done by is also equal to the new expression.

Credits
I need to give some thanks to TechKon for building the basic E* notation. This wouldn't be possible without him. I'd also like to shoutout Jonathan Bowers and Chris Bird, the geniuses behind the original array notation. They practically invented modern googology with their arrays, and I decided to take some inspiration from that for this extension.