The extended hyperoperators need better names. Bowers' extensions are mostly fine, but after that, the only names people use are Aarex's, which are strange to say the least. Let's make that better
We start off with Bowers' expansion. We'll tweak this slightly
{{1}} = expansion
Next is multiexpansion, but that doesn't make sense. It's a 2 in the brackets, just like tetration, so I'll call it...
{{2}} = tetroexpansion
Next up, pentation.
{{3}} = pentoexpansion
Keep the pattern going...
{{4}} = hexoexpansion
{{5}} = heptoexpansion
{{6}} = octoexpansion
{{7}} = ennoexpansion
{{8}} = dekoexpansion
Explosion is next, having 3 brackets.
{{{1}}} = explosion
We'll use the same pattern as we did with expansion
{{{2}}} = tetroexplosion
{{{3}}} = pentoexplosion
...
{{{8}}} = dekoexplosion
Next is detonation...
{{{{1}}}} = detonation
{{{{2}}}} = tetrodetonation
{{{{3}}}} = pentodetonation
....
{{{{8}}}} = dekodetonation
Now, let's generalize. I like the way the expansion-explosion-detonation series is generalized, so I'll keep it.
{{{{{1}}}}} = pentonation
{{{{{{1}}}}}} = hexonation
{{{{{{{1}}}}}}} = heptonation
{{{{{{{{1}}}}}}}} = octonation
{{{{{{{{{1}}}}}}}}} = ennonation
{{{{{{{{{{1}}}}}}}}}} = deconation
How shall we notate the next one? I'll use BEAF. I'll leave out the a,b, so {1,1,2} is really {a,b,1,1,2} where a and b are the arguments
Aarex calls the next one megotion, which I think is good.
{1,1,2} = megotion
I'll keep using the SI prefixes.
{1,1,3} = gigotion
{1,1,4} = terotion
{1,1,5} = petotion
{1,1,6} = exotion
{1,1,7} = zettotion
{1,1,8} = yottotion
{1,1,9} = ronnotion
{1,1,10} = quettotion
Next, we have {1,1,1,2}. Notice how the original hyper-operators have 3 arguments in the array, expansion has 4, and megotion 5? Let's diagonalize our names over this, to finish up the extended hyper-operators. I'll use the original Greek prefixes.
{1,1,1,2} = extotion
{1,1,1,1,2} = eptotion
{1,1,1,1,1,2} = ogdotion
{1,1,1,1,1,1,2} = entotion
{1,1,1,1,1,1,1,2} = dekatotion
And with that, these names are complete.