We examine the minimum entropy coupling problem, where one must find the minimum entropy variable that has a given set of distributions $S = \{p_1, \dots, p_m \}$ as its marginals. Although this problem is NP-Hard, previous works have proposed algorithms with varying approximation guarantees. In this paper, we show that the greedy coupling algorithm of [Kocaoglu et al., AAAI'17] is always within $\log_2(e)$ ($\approx 1.44$) bits of the minimum entropy coupling. In doing so, we show that the entropy of the greedy coupling is upper-bounded by $H\left(\bigwedge S \right) + \log_2(e)$. This improves the previously best known approximation guarantee of $2$ bits within the optimal [Li, IEEE Trans. Inf. Theory '21]. Moreover, we show our analysis is tight by proving there is no algorithm whose entropy is upper-bounded by $H\left(\bigwedge S \right) + c$ for any constant $c<\log_2(e)$. Additionally, we examine a special class of instances where the greedy coupling algorithm is exactly optimal.