The Ladder of Exponents

A student experiments with the sequence $a_1 = x$, $a_{n+1} = x^{a_n}$ for a positive number $x$ — in effect building the infinite exponent tower $x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$. She finds a value of $x$ for which the sequence converges to exactly $2$. What is $x$?