Here's an example making Gerry's suggestion explicit (I was curious what an example would look like): $a = {\omega}2^{1/3}$, $b = 2^{1/6}$, where $\omega$ is a nontrivial cube root of unity. By PARI, $a + b$ has minimal polynomial
$$
x^{12} - 8x^9 + 18x^8 + 12x^7 + 20x^6 - 72x^5 + 276x^4 - 232x^3 + 180x^2 + 24x + 4.
$$

Without giving it a lot of thought, you might think the minimal polynomial might be something like
$$
((x-2^{1/3})^6-2)((x - {\omega}2^{1/3})^6-2)((x - {\omega}^22^{1/3})^6-2)
$$
and this polynomial of degree 18 does have rational coefficients, but it factors as
$$
(x^6 - 4x^3 - 18x^2 - 12x + 2)
(x^{12} - 8x^9 + 18x^8 + 12x^7 + 20x^6 - 72x^5 + 276x^4 - 232x^3 + 180x^2 + 24x + 4).
$$
Yes, the second factor is the polynomial above. The multiplicative relations among
$a$ and $b$ account for such breaking up.