Summary: We prove global existence of the solutions of the loglog energy-supercritical Klein-Gordon equation \(\partial_{tt} u - \triangle u + u = -|u|^{\frac{4}{n - 2}} u \log^{\gamma} \Big(\log(10 + |u|^2) \Big)\), with \(n \in \{3, 4, 5\}\), \(0 < \gamma < \gamma_n\), and data \((u_0, u_1) \in H^k \times H^{k - 1}\) for \(k > 1\) (resp. \(\frac{7}{3} > k > 1\)) \(if n \in \{3, 4\}\) (resp. \(n = 5\)). The proof is by contradiction. Assuming that blow-up occurs at a maximal time of existence, we perform an analysis close to this time in order to find a finite bound of a Strichartz-type norm, which eventually leads to a contradiction with the blow-up assumption.