= Solution
With the usual exterior-boundary interpretation that the <harmonic function> remains bounded as $r\to\infty$, positive radial powers and $\log r$ are excluded. Hence
$$
\boxed{\phi(r,\theta)=A_0+\sum_{n=1}^{\infty}r^{-n}(C_n\cos n\theta+D_n\sin n\theta).}
$$
The <series> must converge locally for $r>1$, with any required finite boundary values at $r=1$. These are the interior <harmonic Fourier expansions in planar concentric domains> after the inversion $r\mapsto1/r$.
If “finite” means only finite at each finite point, without a condition at infinity, growing solutions such as $r\cos\theta$ and $\log r$ are allowed. Under that literal reading the general exterior expansion is
$$
A_0+B_0\log r+\sum_{n\ge1}\big[(A_nr^n+C_nr^{-n})\cos n\theta+(B_nr^n+D_nr^{-n})\sin n\theta\big],
$$
with local convergence. \b[Boundedness at infinity is the condition selecting the decaying exterior expansion.]
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