The answer is, indeed, negative. WLOG $v(0)=0$. Take the intersection of the region $-A<v<2A$ with a huge disk (more precisely, take the connected component $\Omega$ of this intersection containing the origin). It is simply connected by the maximum principle. The nice thing about the plane is that once we have a curve on the boundary that passes not too far from the origin, we can make the harmonic measure of the circle piece of the boundary arbitrarily small by choosing big enough radius. Now, the harmonic measures of the pieces $v=-A$ and $v=2A$ are balanced essentially as $2:1$, so the piece $v=2A$ has harmonic measure about $1/3$. Also $u$ is at least $-A^3$ everywhere in $\Omega$ and at least $8A^3$ on the piece $v=2A$. So, $u(0)\ge -\frac 23 A^3+\frac 13 8A^3=2A^3$. Since it is true for all $A>0$, $u(0)=+\infty$.