This evening next idea visited me: why not to write program for the automatical generation of names of large numbers.
Here I defined rules for my system of names of numbers.
Then I wrote in Pascal the program, which generates names of numbers, according this system, and also the program writes expressions in Latex for each number.
program googologisms;
var a:array[0..20] of string;
b:array[0..4] of string;
i,j,k:integer;
st1,st2,st3,st4, st5,st6,st7,st8:string;
begin
a[1]:='un'; a[2]:='b';
a[3]:='tr'; a[4]:='quadr';
a[5]:='quint'; a[6]:='sext';
a[7]:='sept'; a[8]:='oct';
a[9]:='non'; a[10]:='dek';
a[11]:='unun'; a[12]:='bun';
a[13]:='trun'; a[14]:='quadrun';
a[15]:='quintun'; a[16]:='sextun';
a[17]:='septun'; a[18]:='octun';
a[19]:='nonun'; a[20]:='vig';
b[0]:='om'; b[1]:='add';
b[2]:='ult'; b[3]:='ex';
for k:=1 to 20 do begin
write('= NEW CYCLE W^k HAS BEGUN ='); writeln(' ');writeln('== ==');
for j:=1 to 20 do begin
writeln('== NEW CYCLE W.j HAS BEGUN =='); writeln;
for i:=1 to 20 do begin
st1:=a[i]+b[1]; st2:='-'+a[j]+b[2];
st3:='-'+a[k]+b[3]; st4:=b[0]; st5:=' <math>f_{\omega';
st6:='^{'+k+'}';st7:='.'+j; st8:='+'+i+'}(10)</math>';
if k=1 then st3:=''; if k=1 then st6:='';
if j=1 then st2:=''; if j=1 then st7:='';
write(st1,st2,st3,st4);
writeln(st5,st6,st7,st8); writeln;
end; end; end;
end.
As you can see it is very simply: three nested 20-step-cycles are working for expression <math>f_{\omega^{k}.j+i}(10)</math>.
Below are about 8000 names of numbers with expressions in Latex. They were not written by human, but by this program during just several seconds.
This way, using rules of mentioned system of names as well as using more complex FGH-expressions and more than three nested cycles it is very simply to generate and to write any number of names of large numbers
result of program working (about 8 000 names and expressions) you can see below (just click links):
10 august 2016
P.S. Later I have written another one program for generation of googologisms before <math>\omega</math>. Results of the working of this program you can see here.