Can the product of three complex numbers ever be real?

For example $z^3=1$, where $z\neq1.$

Id est, $$a=b=c=-\frac{1}{2}+\frac{\sqrt3}{2}i.$$

I'm not sure I understood your question, but I suppose that the equality$$i\times(1+i)\times(1+i)=-2$$answers it.

If you represent a complex number using polar coordinates (angle and a distance from zero), it is known that multiplying the numbers in this trigonometric form is way easier than in the algebraic form - you simply multiply the distance and add the angles:

$$z_1=r_1(\cos(ϕ_1)+i\sin(ϕ_1))$$ $$z_2=r_2(\cos(ϕ_2)+i\sin(ϕ_2))$$ $$z_1z_2=r_1r_2(\cos(ϕ_1+ϕ_2)+i\sin(ϕ_1+ϕ_2))$$

Once you are accustomed to this, the rest is simple. If $ϕ$ is parallel with the x axis (0 or 180°, $\sin ϕ=0$), the number is real, and so your only task is to find three angles that add up to 0 (mod 180°). There is an infinite number of them.