MathNet 2015-part-02 — 191 problems (EN–VI) Question 74

Let $a = \sqrt[2015]{2015}$ and $(a_n)$ be the sequence such that $a_1 = a$ and $a_{n+1} = a^{a_n}$ for $n \ge 1$. Does there exist a positive integer $n$ such that $a_n \ge 2015$?