Such sequences are called sum free sequences. In the paper "On a question about sum-free sequences", Deshouillers, Erdős and Melfi construct a sequence where $a_n$ is $o(n^{3+\epsilon})$. Luczak and Schoen, later improved this to a construction of a sum free sequence with $a_n=o(n^{2+\epsilon})$, and show that the exponent $2$ is minimal.

I should mention that the construction, referenced above is related to finding suitable perturbations starting from the set of cubes of integers. Of course for the purposes of the particular question in the OP, one can be more explicit at the cost of not being optimal. For example we can define a sequence $\{a_n\}$ as
$$a_1=2 \, ,\quad a_{2n}=a_{2n-1}+1\, , \quad a_{2n+1}=1+\sum_{i=1}^{2n}a_i$$
which is obviously a sum free sequence and grows as $\sim (\sqrt{3})^n$.