Motivated by the problem of bounding the number of rays of plane tropical curves we study the following question: Given $n\in\mathbb{N}$ and a unimodular $2$-simplex $Δ$ what is the maximal number of vertices a lattice polytope contained in $n\cdot Δ$ can have? We determine this number for an infinite subset of $\mathbb{N}$ by providing a family of vertex-maximal polytopes and give bounds for the other cases.