If a function $\Phi$ satisfy
$$ \nabla^2 \Phi(r, \theta, \phi) = 0 $$
then the general solution can be written as
$$ \Phi(r, \theta, \phi) = \sum_{l=0}^\infty \sum_{m = -l}^l (A_l r^l + B_l r^{-l-1}) P_l^m(\cos\theta) e^{\pm i m \phi} $$
Where $A_l$ and $B_l$ are constants and $P_l^m$ is the associated Legendre polynomial.
Can I use
$$ \sum_{l=0}^\infty \sum_{m = -l}^l (A_l r^l + B_l r^{-l-1}) P_l^m(\cos\theta) e^{\pm i m \phi} $$
to express arbitrary functions including the functions even not a solution of the Laplace equation?
As the first comment below the question, the answer is no. For all valid pair of $l$ and $m$ and any constant $A$,
$$ A r^l P_l^m(\cos\theta) e^{\pm i m \phi} $$
and
$$ A r^{-l-1} P_l^m(\cos\theta) e^{\pm i m \phi} $$
are the solutions of the Laplace equation
$$ \nabla^2 \Phi(r, \theta, \phi) = 0 $$
This mean if a function $f(r, \theta, \phi)$ can be express as
$$ f(r, \theta, \phi) = \sum_{l=0}^\infty \sum_{m = -l}^l (A_l r^l + B_l r^{-l-1}) P_l^m(\cos\theta) e^{\pm i m \phi} $$
, $f$ must be a solution of the Laplace equation. To compose a function cannot be expressed as $\sum_{l=0}^\infty \sum_{m = -l}^l (A_l r^l + B_l r^{-l-1}) P_l^m(\cos\theta) e^{\pm i m \phi}$ is trivial.
$$ A r^{l-1} P_l^m(\cos\theta) e^{\pm i m \phi} $$
is an example.
About the Green's function
The Green's function of 3-D Laplace operator is
$$ G(\vec{x}, \vec{x'}) = 4\pi \sum^\infty_{l = 0} \sum^l_{m = -l} \frac{1}{(2l + 1)}\frac{r^l_{\lt}}{r^{l + 1}_{\gt}} Y^*_{lm}(\theta', \phi')Y_{lm}(\theta, \phi)\frac{\left[ 1 - \left( \dfrac{a}{r_\lt} \right)^{2l + 1} \right]\left[ 1 - \left( \dfrac{r_\gt}{b} \right)^{2l + 1} \right] }{1 - \left( \dfrac{a}{b} \right)^{2l + 1}} $$
$\vec{x}$ and ${x'}$ are $(r, \theta, \phi)$ and $(r', \theta', \phi')$ correspondingly. $r = a$ is the inner boundary surface, and $r = b$ is the outer boundary surface. You can read $G(\vec{x}, \vec{x'})$ as $G(r, \theta, \phi;\ r', \theta', \phi')$. When $|\vec{x}| \lt |\vec{x'}|$, $r_{\lt} = |\vec{x}| = r$ and $r_{\gt} = |\vec{x'}| = r'$. When $|\vec{x}| \ge |\vec{x'}|$, $r_{\gt} = |\vec{x}| = r$ and $r_{\lt} = |\vec{x'}| = r'$.
$$\begin{align*} \begin{cases} r_{\lt} = |\vec{x}| = r \\ r_{\gt} = |\vec{x'}| = r' \end{cases} && \text{for } |\vec{x}| \lt |\vec{x'}| \\ \begin{cases} r_{\lt} = |\vec{x'}| = r' \\ r_{\gt} = |\vec{x}| = r \end{cases} && \text{for } |\vec{x}| \ge |\vec{x'}| \\ \end{align*} $$
For case that $|\vec{x}| \lt |\vec{x'}|$,
$$\begin{align*} G(\vec{x}, \vec{x'}) &= 4\pi \sum^\infty_{l = 0} \sum^l_{m = -l} \frac{1}{(2l + 1)} \frac{r^l_{\lt}}{r^{l + 1}_{\gt}} Y^*_{lm}(\theta', \phi') Y_{lm}(\theta, \phi) \frac{ \left[ 1 - \left( \dfrac{a}{r_\lt} \right)^{2l + 1} \right] \left[ 1 - \left( \dfrac{r_\gt}{b} \right)^{2l + 1} \right] }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } \\ &= 4\pi \sum^\infty_{l = 0} \sum^l_{m = -l} \frac{1}{(2l + 1)} \frac{r^l}{r'^{l + 1}} Y^*_{lm}(\theta', \phi') Y_{lm}(\theta, \phi) \frac{ \left[ 1 - \left( \dfrac{a}{r} \right)^{2l + 1} \right] \left[ 1 - \left( \dfrac{r'}{b} \right)^{2l + 1} \right] }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } && (r_\lt = r,\ r_\gt = r') \\ &= \sum^\infty_{l = 0} \sum^l_{m = -l} \frac{4 \pi}{(2l + 1)} \frac{1}{r'^{l + 1}} Y^*_{lm}(\theta', \phi') \frac{ \left[ 1 - \left( \dfrac{r'}{b} \right)^{2l + 1} \right] }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } r^l \left[ 1 - \left( \dfrac{a}{r} \right)^{2l + 1} \right] Y_{lm}(\theta, \phi) && (\text{rearrange variables}) \\ &= \sum^\infty_{l = 0} \sum^l_{m = -l} \frac{4 \pi}{(2l + 1)} \frac{ Y^*_{lm}(\theta', \phi') }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } (-b^{-2l - 1} r'^l + r'^{- l - 1}) (r^l - a^{2l + 1} r^{-l - 1}) Y_{lm}(\theta, \phi) && (\text{rearrange variables}) \\ &= \sum^\infty_{l = 0} \sum^l_{m = -l} C_{lm} (-b^{-2l - 1} r'^l + r'^{- l - 1}) (r^l - a^{2l + 1} r^{-l - 1}) Y_{lm}(\theta, \phi) && ( \text{let } C_{lm} = \frac{4 \pi}{(2l + 1)} \frac{ Y^*_{lm}(\theta', \phi') }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } ) \\ \end{align*}$$
We got $A_l = C_{lm}(-b^{-2l - 1} r'^l + r'^{- l - 1})$ and $B_l = -a^{2l + 1} C_{lm}(-b^{-2l - 1} r'^l + r'^{- l - 1})$
For case that $|\vec{x}| \ge |\vec{x'}|$,
$$\begin{align*} G(\vec{x}, \vec{x'}) &= 4\pi \sum^\infty_{l = 0} \sum^l_{m = -l} \frac{1}{(2l + 1)} \frac{r^l_{\lt}}{r^{l + 1}_{\gt}} Y^*_{lm}(\theta', \phi') Y_{lm}(\theta, \phi) \frac{ \left[ 1 - \left( \dfrac{a}{r_\lt} \right)^{2l + 1} \right] \left[ 1 - \left( \dfrac{r_\gt}{b} \right)^{2l + 1} \right] }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } \\ &= 4\pi \sum^\infty_{l = 0} \sum^l_{m = -l} \frac{1}{(2l + 1)} \frac{r'^l}{r^{l + 1}} Y^*_{lm}(\theta', \phi') Y_{lm}(\theta, \phi) \frac{ \left[ 1 - \left( \dfrac{a}{r'} \right)^{2l + 1} \right] \left[ 1 - \left( \dfrac{r}{b} \right)^{2l + 1} \right] }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } && (r_\lt = r',\ r_\gt = r) \\ &= \sum^\infty_{l = 0} \sum^l_{m = -l} \frac{4 \pi}{(2l + 1)} r'^l Y^*_{lm}(\theta', \phi') \frac{ \left[ 1 - \left( \dfrac{a}{r'} \right)^{2l + 1} \right] }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } r^{-l - 1} \left[ 1 - \left( \dfrac{r}{b} \right)^{2l + 1} \right] Y_{lm}(\theta, \phi) && (\text{rearrange variables}) \\ &= \sum^\infty_{l = 0} \sum^l_{m = -l} \frac{4 \pi}{(2l + 1)} \frac{ Y^*_{lm}(\theta', \phi') }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } (r'^l - a^{2l + 1} r'^{- l - 1}) (-b^{-2l - 1} r^l + r^{-l - 1}) Y_{lm}(\theta, \phi) && (\text{rearrange variables}) \\ &= \sum^\infty_{l = 0} \sum^l_{m = -l} C_{lm} (r'^l - a^{2l + 1} r'^{- l - 1}) (-b^{-2l - 1} r^l + r^{-l - 1}) Y_{lm}(\theta, \phi) && ( \text{let } C_{lm} = \frac{4 \pi}{(2l + 1)} \frac{ Y^*_{lm}(\theta', \phi') }{ 1 - \left( \dfrac{a}{b} \right)^{2l + 1} } ) \\ \end{align*}$$
We got $A_l = -b^{-2l - 1} C_{lm} (r'^l - a^{2l + 1} r'^{- l - 1})$ and $B_l = C_{lm} (r'^l - a^{2l + 1} r'^{- l - 1})$
After comparing $A$ and $B$ from the two cases, we find that only when
$$ \frac{a^{2l + 1}}{b^{2l + 1}} = 1 \Rightarrow a = b $$
$G(\vec{x}, \vec{x'})$ can be expressed as $\sum_{l=0}^\infty \sum_{m = -l}^l (A_l r^l + B_l r^{-l-1}) P_l^m(\cos\theta) e^{\pm i m \phi}$. However, the interest area also becomes zero when $a = b$ and the Green's function becomes meaningless. If $a \lt b$, $A_l$ and $B_l$ will be functions of $r$ and have a jump at $r'$.