Exponential factorial

The exponential factorial is an exponential version of the factorial, recursively defined as $$a_0 = 1$$ and $$a_n = a^{a_{n - 1}}$$. For example, $$a_6 = 6^{5^{4^{3^{2^1}}}}$$.

The first few $$a_n$$ for $$n = 0$$, 1, 2, 3, ... are 1, 1, 2, 9, 262144, etc. The exponential factorial of five has 183231 digits. The sum of the reciprocals of these numbers is highly unusual, namely $$1.611114925808376736\underbrace{111\ldots111}_{183213}272243682859\ldots$$