Why not make a few names with the notation then? (well, if few means hundreds, yes.)
And We'll start with a classic....
Gigafugathree = T[3,3] = 3^^^3 = 3^^7,625,597,484,987
(also known as Tritri)
Terafugathree = T[3,4] = 3^^^^3 = G1
(also known as Grahal)
Petafugathree = T[3,5]
(used to be known as Great Graham as a misunderstanding of Grahal vs Graham)
Exafugathree = T[3,6]
Zettafugathree = T[3,7]
Yottafugathree = T[3,8]
Ronnafugathree = T[3,9]
Quettafugathree = T[3,10]
Thritrihect = T[3,100]
Thritrikill = T[3,1000]
Thritrimeg = T[3,1000000]
Thritrigig = T[3,1000000000]
Thritriter = T[3,1000000000000]
Thritrimegafuge = T[3,T[3,2]] = {3,3,{3,3,2}}
Thritrigigafuge = T[3,T[3,3]] = {3,3,{3,3,3}}
Thritriterafuge = T[3,T[3,4]] = {3,3,{3,3,4}}
Thritripetafuge = T[3,T[3,5]] = {3,3,{3,3,5}}
Thritri-exafuge = T[3,T[3,6]] = {3,3,{3,3,6}}
Thritrizettafuge = T[3,T[3,7]] = {3,3,{3,3,7}}
Thritriyottafuge = T[3,T[3,8]] = {3,3,{3,3,8}}
Thritrironnafuge = T[3,T[3,9]] = {3,3,{3,3,9}}
Thritriquettafuge = T[3,T[3,10]] = {3,3,{3,3,10}}
Thritriaduplex = T[3,T[3,T[3,3]]] = {3,3,{3,3,{3,3,3}}}
Thritriatriplex = T[3,T[3,T[3,T[3,3]]]] = T[5,1,2] = {3,5,1,2}
Thritriaquadriplex = T[6,1,2]
Thritriaquintiplex = T[7,1,2]
Thritriasextiplex = T[8,1,2]
Thritriaseptiplex = T[9,1,2]
Thritrioctaplex = T[10,1,2]
Going back a few steps and coming back again much faster we have:
Thrigugolda = T[3,0,2]* = T[3,3]
Thrigraatada = T[3,1,2] = T[3,T[3,3]]
Thrigreeda = T[3,2,2] = {3,3,2,2} = {3,{3,2,2,2},1,2} = {3,{3,3,1,2},1,2}
Thrigrinninda = T[3,3,2] = {3,3,3,2}
(also called Thrigugolthra, will get to that)
Thrigolaada = T[3,4,2] = {3,3,4,2}
Thrigruelohda = T[3,5,2] = {3,3,5,2}
Thrigaspda = T[3,6,2] = {3,3,6,2}
Thriginorda = T[3,7,2] = {3,3,7,2}
Thrigarganda = T[3,8,2] = {3,3,8,2}
Thrigoogonda = T[3,9,2] = {3,3,9,2}
Thrigoogoada = T[3,10,2] = {3,3,10,2}
Thrigugolthra = T[2,1,3] = {3,2,1,3} = {3,3,3,2}
(also Thrigrinninda)
Thrigugolthra-plexitris = T[3,1,3]
Thrigugolthra-duplexitris = T[4,1,3]
Thrigugolthra-triplexitris = T[5,1,3]
Thrigugolthra-quadriplexitris = T[6,1,3]
Thrigugolthra-quintiplexitris = T[7,1,3]
Thrigugolthra-sextiplexitris = T[8,1,3]
Thrigugolthra-septiplexitris = T[9,1,3]
Thrigugolthra-octiplexitris = T[10,1,3]
Thrigugoltesla = T[3,3,3] = {3,3,3,3}
(Bowers himself calls this tetratri, and this can also be Thrithroogol)
Thrigugolpenta = T[3,3,4] = {3,3,3,4}
Thrigugolhexa = T[3,3,5] = {3,3,3,5}
Thrigugolhepta = T[3,3,6] = {3,3,3,6}
Thrigugolocta = T[3,3,7] = {3,3,3,7}
Thrigugolenna = T[3,3,8] = {3,3,3,8}
Thrigugoldeka = T[3,3,9] = {3,3,3,9}
Thrigugolendeka = T[3,3,10] = {3,3,3,10}
Grand Thrithroogol = T[2,1,1,2]
Thrithrangol = T[3,1,1,2]
(because of the annoying 3 base we have this is the same as Thrigugold)
Thrithroogugoldasuplex / Thrithroograatada = T[3,2,1,2]
Thrithroogreeda / Thrithroogugolthra = T[3,3,1,2]
Thrithroogugolthra-plexitris / Thrithroograatathra = T[3,1,2,2]
(again annoying base 3 problem makes for slightly lower levels..., lets get fast now)
Thrithroogugoltesla = T[3,3,3,2] = {3,3,3,3,2}
Thrithroogugolpeta = T[3,3,3,3] = {3,3,3,3,3}
(also pentatri by Bowers, and also...)
Thritetroogol = T[3,3,3,3] = T[5,,2]
Thripentoogol = T[3,3,3,3,3] = T[6,,2]
Thrihexoogol = T[3,3,3,3,3,3] = T[7,,2]
Thriheptoogol = T[3,3,3,3,3,3,3] = T[8,,2]
Thri-ogdoogol = T[3,3,3,3,3,3,3,3] = T[9,,2]
Thri-entoogol = T[3,3,3,3,3,3,3,3,3] = T[10,,2]
Thrigrangahlah = T[3,2,,2] = {3,3,2(1)2} (also dupertri)
Thrigreagahlah = T[3,3,,2] = {3,3,3(1)2} = {3,3(1)3} = T[3,,3] (also latri, we also have Thrigugoldgahlah, and Thrigotrigahlah)
(because of all of these weird stuff with having base 3, we will have to skip to...)
Thrigotergahlah = T[3,,4]
Thrigoppegahlah = T[3,,5]
Thrigohexgahlah = T[3,,6]
Thrigohepgahlah = T[3,,7]
Thrigo-ahtgahlah = T[3,,8]
Thrigo-enngahlah = T[3,,9]
Thrigodekahlah = T[3,,10]
Using the weird stuff in our notation we can have
Thrigrangoldgahlah = T[3,,1,2] (also Grand Thrigodgoldgahlah)
Thrigreagoldgahlah = T[3,2,,1,2] (also Grand Thrigrangoldgahlah)
Thrigigangoldgahlah = T[3,3,,1,2] = T[3,,2,2] (That implies it's also Thrigugold-carta-godgoldgahlah)
Thrithroogol-carta-godgoldgahlah = T[3,3,3,,1,2] = T[4,,2,2]
Thritetroogol-carta-godgoldgahlah = T[3,3,3,3,,1,2] = T[5,,2,2]
Thrigrantrigoldgahlah = T[3,2,,2,2]
Thrigreatrigoldgahlah = T[3,3,,2,2] = T[3,,3,2] (also Thrigugoldtrigoldgahlah)
Thrigodpeggoldgahlah = T[3,,4,2]
Thrigodekoldgahlah = T[3,,9,2]
Thrigrandeutergoldgahlah = T[3,,1,3]
Thrigotritogoldgahlah = T[3,,3,3] = T[3,,0,,2]
Thrigotetertogoldgahlah = T[3,,3,3,3] = T[4,,0,,2]
Thrigodpeptogoldgahlah = T[3,,3,3,3,3] = T[5,,0,,2]
Thrigodektogoldgahlah = T[10,,0,,2]
Thrideutergrangahlah = T[3,,1,,2]
<BEYOND THIS THE NOTATION IS CURRENTLY NOT FORMALIZED YET> <BE READY FOR SURPRISES>
Thrideutergotrigahlah = T[3,,2,,2] -> (#^#*#^#)+(#^#*#^#)
Thrideutergodgoldgahlah = T[3,,3,,2] -> #^#*#^#*#
Thrideutergrangoldgahlah = T[3,,1,2,,2] -> (#^#*#^#*#)+#
Thrigodgahlah-carta-deutero-godgoldgahlah = T[3,,2,2,,2] -> (#^#*#^#*#)+#^#
Thrigrangoldgahlah-carta-deutero-godgoldgahlah = T[3,,1,3,,2] -> (#^#*#^#*#)+(#^#*#)+#
Thrigrandeutergoldgahlah-carta-deutero-godgoldgahlah = T[3,,1,1,2,,2] -> (#^#*#^#*#)+(#^#*##)+#
Thrigrantritogoldgahlah-carta-deutero-godgoldgahlah = T[3,,1,1,1,2,,2] -> (#^#*#^#*#)+(#^#*###)+#
Thrideutergrangahlah-carta-deutero-godgoldgahlah = T[3,,1,,3] -> (#^#*#^#*#)+(#^#*#^#)+#
Thrideutergotrigoldgahlah = T[3,,2,,3] -> #^#*#^#*#*3
Thrideutergodeutergoldgahlah = T[3,,3,,3] -> #^#*#^#*##
Thrideutergrandeutergoldgahlah = T[3,,1,2,,3] -> (#^#*#^#*##)+#
Thrideutero-godgahlah-carta-godeutergoldgahlah = T[3,,2,2,,3] -> (#^#*#^#*##)+(#^#)
Thrideutero-grangoldgahlah-carta-godeutergoldgahlah = T[3,,1,3,,3] -> (#^#*#^#*##)+(#^#*#)+#
These names are so tricky, let's skip them for now
T[3,,1,1,2,,3] -> (#^#*#^#*##)+(#^#*##)+#
T[3,,1,1,1,2,,3] -> (#^#*#^#*##)+(#^#*###)+#
T[3,,1,,4] -> (#^#*#^#*##)+(#^#*#^#)+#
...
Seeing this, maybe T[3,,3,,n] is a better fundamental sequence than T[3,,1,,n].
NOW LET'S ACCELERATE TOWARDS THE UNKNOWN!!!
T[4,,,2] = T[3,,3,,3] = Thrideutergodeutergoldgahlah
T[3,2,,,2]
T[3,3,,,2]
T[3,,3,,,2]
T[3,,,3]
T[3,,,1,2]
T[3,,,3,,,3]
T[3,2,,,,2]
T[3,,,,1,2]
T[3,,,,3,,,,3]
T[3,2,,,,,2]
T[3,,,,,1,2]
T[3,,,,,3,,,,,3]
T[3,,,,,,,2]
T[3,{8}2]
T[3,{27}2]
T[3,{1,2}2]
the comma is important!!
T[3,{2,2}2] = T[3,{1,2}3]
T[3,{3,3}2]
T[3,{1,1,1,1,1,1,2}2]
T[3,{a,,2}2] should have ~ #^#^#^# growth -- we are getting somewhere!
T[3,{3,{a,,2}2}2] should have ~ #^#^#^#^#^# growth...
NOW WE ARE STUCK AT #^^# level, just like BEAF, and we need a better notation, well at least we got past most notations, including Conway's Chained Arrow Notation (uh forget that it doesn't go much further than graham's level) and even extensions of it like HECCAN, and passed the one-row limit for BM4=BM2.3, and also the well-definedness of BEAF.
Well, more is for next time :)