Multiplication of Power Tower

Multiply the equation by $(2^{2^0}-1)$ . (Take note that $2^{2^0}-1=1$ so it won't change the value of equation)

The equation now becomes

$\left( { 2 }^{ { 2 }^{ 0 } }-1 \right) \left( { 2 }^{ { 2 }^{ 0 } }+1 \right) \left( { 2 }^{ { 2 }^{ 1 } }+1 \right) \left( { 2 }^{ { 2 }^{ 2 } }+1 \right) ...\left( { 2 }^{ { 2 }^{ 2015 } }+1 \right)$

Take note that

$(2^a-1)(2^a+1)=2^{2a}-1$

$2^{{2^k} \times 2}=2^{2^{k+1}}$

Using these two identities,

$\left( { 2 }^{ { 2 }^{ 0 } }-1 \right) \left( { 2 }^{ { 2 }^{ 0 } }+1 \right) \left( { 2 }^{ { 2 }^{ 1 } }+1 \right) \left( { 2 }^{ { 2 }^{ 2 } }+1 \right) ...\left( { 2 }^{ { 2 }^{ 2015 } }+1 \right)$

= $\left( { 2 }^{ { 2 }^{ 1 } }-1 \right) \left( { 2 }^{ { 2 }^{ 1 } }+1 \right) \left( { 2 }^{ { 2 }^{ 2 } }+1 \right)...\left( { 2 }^{ { 2 }^{ 2015 } }+1 \right)$

= $\left( { 2 }^{ { 2 }^{ 2 } }-1 \right) \left( { 2 }^{ { 2 }^{ 2 } }+1 \right) ...\left( { 2 }^{ { 2 }^{ 2015 } }+1 \right)$

.

.

.

= $\left( { 2 }^{ { 2 }^{ 2015 } }-1 \right) \left( { 2 }^{ { 2 }^{ 2015 } }+1 \right)$

= ${2}^{{2}^{2016}}-1$

$A=2^{2016}$

$log_{2}{A} = \boxed{2016}$