User blog comment:Pellucidar12/Attempt at a FGH related notation/@comment-28606698-20170415185346/@comment-30754445-20170416084133

"Also with the 5th term it can go into uncomputable numbers (if it works the way I intended it to). "

By the very definition of "uncomputable", there is no way to reach uncomputable numbers by any recursive definition.

So either your notation would end up weaker than you intended, or it will have gaps.

And speaking of gaps and limitations:

If you allow only numbers inside your arrays and require an intuitive ordering (e.g. {7,1,3} must be bigger than {7,1,2} and longer arrays must represent bigger ordinals) then the theoretical limit will be much lower:  ωω.

Why? Because ωω is the ordinal you get by listing all number-arrays in that ordering. Just like you can't have an integer between "1" and "2", you cannot squeeze in an array between (say) {7,1,2} and {7,1,3}.

That is - by the way - the reason that Linear Bowers Arrays (which seem to be similar in concept to your own notation) have strength of ωω.

the order-type of all such arrays is ωω.