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This is a subspecies of BEAF.

I have wanted an easy notation that I can understand.

I think the easiest way to make a big number is "piling up."

So, I will make a piling-up notation.

PU(a,b)=a^b

PU(a,b,c)=a\uparrow^c b =PU(a,PU(a,b-1,c),c-1)

PU((anything),1)=PU((anything))

Now, you need to understand that the default of PU is 1.

In math language, PU(X)=PU(X,1)=PU(X,1,1)=\cdots

I piled up b's. What's next? Piling up c's. Therefore,

PU(a,b,1,2)=PU(a,b,PU(a,b-1,1,2),1)

PU(a,1,X)=a

Then, PU(a,b,1,2)\simeq a\rightarrow b\rightarrow b\rightarrow 2

Next, I can think the same thing:

PU(a,b,c,2)=PU(a,PU(a,b-1,c,2),c-1,2)\simeq a\rightarrow b\rightarrow b\rightarrow c

(note: this approximation is just shows level. Since I know chain notation, I can think (how big PU is) better.)

Now, I will jump to polynominal. K means some one's.

PU(a,b,c,d,\cdots)=PU(a,PU(a,b-1,c,d,\cdots),c-1,d,\cdots)

PU(a,b,K,1,d,\cdots)=PU(a,b,K,PU(a,b-1,K,1,d,\cdots),d-1,\cdots)

PU(a,1,X)=a

For example,

PU(a,b,1,1,2)

=PU(a,b,1,PU(a,b-1,1,1,2),1)

=PU(a,b,1,PU(a,b-1,1,PU(a,b-2,1,\cdots)

I think this is as big as a\rightarrow_2 b\rightarrow_2 2 in extended chain notation, but I don't know how big it is exactly.

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