A necklace that has no rotational symmetry is called a _primitive_ or _aperiodic_ necklace, or a _Lyndon word_ (if written as a word by taking the lex-smallest rotation.) The number of primitive necklaces is given by almost the same formula as the one you give for the number of necklaces, but with Euler's $\phi$ replaced by the Mobius function $\mu$ (see Wikipedia#Aperiodic_necklaces)): $$\frac{1}{n} \sum_{d |n } \mu(d) a^{n/d}.$$
Intuitively, we would expect that for large $n$ a random necklace would be almost surely be primitive, since there would have to be a large number of coincidences to find a rotational symmetry. To prove this, just take the ratio
$$\frac{\frac{1}{n} \sum_{d |n } \mu(d) a^{n/d}}{ \frac{1}{n} \sum_{d |n } \phi(d) a^{n/d}} = \frac{a^n + O(a^{n-1}) }{ a^n + O(a^{n-1}) }$$
since the highest order terms in both polynomials corresponds to $d=1$, and $\mu(d) = \phi(d) = 1$. Then the limit as $n \rightarrow \infty$ is $1$.