Summary: Let \(\{Y_{\mathfrak{B}}(x):x\in\mathfrak{B}\}\) be a discrete Gaussian free field in a two-dimensional box \(\mathfrak{B}\) of side length \(S\) with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex \(x\) is given a weight of \(e^{\gamma Y_{\mathfrak{B}}(x)}\) for some \(\gamma>0\). We show that for sufficiently small but fixed \(\gamma>0\), for any sequence of scales \(\{S_{k}\}\) there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov-Hausdorff sense to a random metric on the unit square in \(\mathbb{R}^{2}\). In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.