Logic

Let $f\left( x \right) = X$ Then,

${X}^{ { { X }^{ { X }^{ X } } }^{ ... } }=2$

${X}^{ { { X }^{ { X }^{ X } } }^{ ... } } = log_{X} 2$

Substuting, ${X}^{ { { X }^{ { X }^{ X } } }^{ ... } }=2$

$2 = log_{X} 2$

$X^2 = 2$

$X=f\left( x \right)= \sqrt {2}$

Substituting the value of X in $f\left( x \right) = X$ ,

$\large {\frac { x }{ \sqrt { 2 } }= \sqrt {2}}$

$\large { x = 2}$

Thus, $g\left( x \right) ={ f\left( x \right) }^{ 2 }+{ x }^{ 2 }$

$g\left( x \right) = (\large{\sqrt { 2 }^2}) + (\large {2^2})$

$\large {g\left( x \right)}= \large{ 2 + 4} = \boxed {\large {6}}$

Let y=2=f(x)^(f(x)^...)...), Then f(x)^y=2 or f(x)^2 = 2, f(x)=sqrt(2). By Substituting we get x=2 and g(x)=6