The unconventional Weyl point with nonlinear dispersion features higher topological charge $|\mathcal{C}|>1$ and multiple topologically protected Fermi arc states at its boundary. As a novel topological state, it has been attracting widespread attention. However, the unconventional Weyl point with $|\mathcal{C}|=3$ has not yet been reported in realistic materials, even though it has been theoretically proposed for more than a decade. In this work, based on first-principles calculations and theoretical analysis, we predict the existing material $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{LiIO}}_{3}$ as the first realistic example with this unconventional Weyl point. Specifically, in the phonon spectra of $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{LiIO}}_{3}$, two Weyl points with $\mathcal{C}=\ensuremath{-}3$, connected by time-reversal symmetry, appear at the neck crossing point of an hourglass-type band, leading to two hourglass charge-3 Weyl phonons. The symmetry protection and the associated novel triple- and sextuple-helicoid surface arc states of the hourglass charge-3 Weyl phonons are revealed. Our results uncover a hidden topological character of $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{LiIO}}_{3}$ and also show that the phonon spectra are a great platform for exploring unconventional topological states.