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For any logarithmic function, $f(x) = \log_{c}{x}, c\neq1$ , the greatest lower bound for the domain is 0.

Which means: $\frac{\log_g{\frac{\log_g{\frac{A}{g}}}{g}}}{g}>0$ .

After isolating $A$ , we get: $A>g^{g+1}\approx12.99999999$

$\therefore A=13$

This solution is for people who have not learned much about calculus yet (like me). If you have better solutions, post it up:

let

$\\ g=2.218654762\\ \log _{ g }{ \frac { \log _{ g }{ \frac { \log _{ g }{ \frac { A }{ g } } }{ g } } }{ g } } =n,$

then,

${ g }^{ { g }^{ { g }^{ n }g }g }g=A$

its a bit hard to see so here it is: (g^((g^((g^n) g)) g))*g=A

Graphing:

$y={ g }^{ { g }^{ { g }^{ x }g }g }g$

it shows that the minimum for y is when x is as negative as possible, therefore, the minimum of A is extremely close to:

${ g }^{ { g }^{ { g }^{ -9*{ 10 }^{ 99 } }g }g }g$

which can be easily calculated with a scientific calculator. This gives

$A\approx 12.99999999$

And since A is an integer, $A=13$