Keep the largest number round 2

The smallest natural number bigger than any 3-digit natural number, 10000

1000000 pesos in dollars, about 42000

3^^3 = 3^27 ~ 7.6x10¹² - first non-degenerative tetration number

Let A (n,k) = nk (i.e. A (5,90)=590
Let B (n,k) = nn...nb (k digits)
Note that 10^n = A (1,B (0,n))

My entry is A (1, B (0, 15)) wich is a billiard in the long-scale system

not a naive extension, because you defined the whole thing yourself. If I come and start saying things ike "F(n)=UN(UN(UN(7n²+9n+5)))", that's a naive extension.
naive extensions can also be applying weak systems on stronger systems, i.e. f(n)=10^FOOT(n)

Let's define the following structure:

- I define a structure "A" composed of n members and noted {a₀,a₁,...,a_n-1,a_n} with a_n the n-th member
- a₀ is called the base, a₁ is the prime.
- The first non-1 member after the prime is the pilot
- The member immediatly before the pilot is the copilot (should be a 1 or the prime)
- All members from the base to the copilot (excluded) are the passengers
- v(A) is the value of the array, and is determined as the following:
-> If the base is 1, v(A) = 1
-> If the prime is 1, v(A) = a₀
-> If there is no pilot (= the pilot is 1), v(A) = a₀^a₁
-> Otherwise, decrease the pilot by 1, set all passengers to the base remplace the copilot by a copy of the original array, but the prime in this array is decreased by 1

More simply:
{a} = a
{a,b} = a^b
{...,1,#} = {...}
{a,b,1,1,...,1,1,c,...} = {a,a,a,a,...,a,{a,b-1,1,1,...,1,1,c,...},c-1,...}
{a,b,c,...} = {a,{a,b-1,c,...},c-1,...}

Example:
{3,3,3}
= {3,{3,2,3},2}
= {3,{3,{3,1,3},2},2}
= {3,{3,3,2},2}
= {3,{3,{3,2,2},1},2}
= {3,{3,{3,{3,1,2},1},1},2}
= {3,{3,{3,3,1},1},2}
= {3,{3,3³,1},2}
= {3,3³³,2}
= {3,{3,3²⁷-1,2},1}
= 3^{3,327-1,2}
= 3^3{3,327-2,2}
etc.
= 3^3...33 with 3³³ 3's
= 3↑↑(3↑27)
= 3↑↑(3↑↑3)
= 3↑↑↑3
More generally, {a,b,c} = a↑↑↑...↑↑↑b with c arrows

4-entries array of the form {a,b,c,2} grows at the same rate as Conway's chained arrow notation
It can be shown that {a,b,c,2} >> a→a→(b-1)→(c+1)
And that {a,b,c,d} >> a→a→a→...→a→a→a→(b-1)→(c+1) with d "→a" blocks

5-entries arrays grows fast as f*ck, now imagine b&p = {p,p,p,...,p,p,p} with b p's...

... but I'll use that later, for now I will restrict myself to {2&100,2,2} = 100¹⁰⁰→2→2 = 100¹⁰⁰↑↑2 = (10¹⁰⁰⁰)¹⁰¹⁰⁰⁰ = 10^1000*101000
I know that it is a big jump, but the game is kinda dying, so...

{10,10,2}, which is usually called the Decker