Australian AMC 2014 - Senior Question 25

Question 25

The sequence $2,2^2,2^{2^2},2^{2^{2^2}},\ldots$ is defined by $a_1=2$ and $a_{n+1}=2^{a_n}$ for all $n\ge1$. What is the first term greater than $1000^{1000}$?

Question illustration

$a_4=2^{2^{2^{2}}}$
$a_5=2^{2^{2^{2^{2}}}}$
$a_6=2^{2^{2^{2^{2^{2}}}}}$
$a_7=2^{2^{2^{2^{2^{2^{2}}}}}}$
$a_8=2^{2^{2^{2^{2^{2^{2^{2}}}}}}}$