See if references in this post by Ragib Zaman are useful.

Specifically, the following theorem by Erdős, says that for an increasing sequence $a_k$ of positive integers, in this case $a_k = 2^{3^k}+1$, such that $\underset{n\to \infty}{\lim \sup}\; a_n^{\frac1{2^n}} = \infty$ and $a_n > n^{1+\epsilon}$ for every $\epsilon > 0$ and $n>n_0(\epsilon)$, then the sum $\sum\limits_{n=1}^\infty \frac1{a_n}$ is an irrational number.

It is easy to check that these conditions are met:
$$
\underset{n\to \infty}{\lim \sup} \left( 2^{3^n}+1 \right)^{\frac1{2^n}} =
\lim_{n \to \infty} 2^{(3/2)^n} = \infty
$$
Also clearly $a_n$ grows faster than $n^{1+\epsilon}$ for any $\epsilon > 0$, thus it follows that $\sum\limits_{n=1}^\infty \frac1{2^{3^n}+1}$ is irrational.