= Solution
There is no allowed <spherical harmonic> with $m=l+1$, so $L_+Y_l^l=0$. The equation $-i\hbar\partial_\phi Y=\hbar lY$ first gives $Y=f(\theta)e^{il\phi}$. Combining the differential expressions for $L_x,L_y$ yields
$$
L_+=\hbar e^{i\phi}(\partial_\theta+i\cot\theta\,\partial_\phi).
$$
Its annihilation of the highest-weight state becomes
$$
f'(\theta)-l\cot\theta\,f(\theta)=0.
$$
Integrating $f'/f=l\cot\theta$ gives $f=C\sin^l\theta$, with the constant solution included for $l=0$. Hence
$$
\boxed{Y_l^l(\theta,\phi)=C\sin^l\theta\,e^{il\phi}.}
$$
The <phase> $e^{il\phi}$ is single valued for <integral> orbital <quantum number> $l$; <normalization> fixes only $|C|$, leaving a conventional constant <phase>.
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