Can context-free grammar generate $a^{2^n}$?
It is well known (and not very difficult to prove) that a context-free language over a unary alphabet $\{a\}$ is regular.

Thus, your question is essentially, "is $\{a^{2^n}:n\in \mathbb N\}$ regular?"
And the answer to that is no (easy to prove using the pumping lemma).