We will derive a general formula for the number of regions. Let the three sets have the number of elements $|X| = x$, $|Y| = y$ and $|Z| = z$, respectively. The first two divide the plane into a total of $(x+1)(y+1)$ regions. Each line of the third set intersects the others at $x+y$ points and is divided into $x+y+1$ parts. Each of them divides one of the already obtained areas into two. Thus we get a general formula:
$$S_{x,y,z} = (x+1)(y+1) + z(x+y+1) = x + y + z + xy + xz + yz + 1.$$
We note that $n = x + y + z$. Furthermore, we have that $3(xy + yz + zx) \le (x + y + z)^2 = n^2$ (equivalent to $(x - y)^2 + (y - z)^2 + (z - x)^2 \ge 0$). So $S_{x,y,z} \le n^2/3 + n + 1$ which is less than 128 for $n = 18$ (actually, $S_{6,6,6} = 127$, i.e. $S_{18} = 127$). Also, $S_{6,6,7} = 140$, so $S_n > 128$ for $n \ge 19$. Therefore, the answer is 18. $\square$