  Assume that $K$ is an algebraically closed field and denote by $KG(R)$ the
Krull-Gabriel dimension of $R$, where $R$ is a locally bounded $K$-category (or
a bound quiver $K$-algebra). Assume that $C$ is a tilted $K$-algebra and
$\widehat{C},\check{C},\widetilde{C}$ are the associated repetitive category,
cluster repetitive category and cluster-tilted algebra, respectively. Our first
result states that $KG(\widetilde{C})=KG(\check{C})\leq KG(\widehat{C})$. Since
the Krull-Gabriel dimensions of tame locally support-finite repetitive
categories are known, we further conclude that
$KG(\widetilde{C})=KG(\check{C})=KG(\widehat{C})\in\{0,2,\infty\}$. Finally, in
the Appendix Grzegorz Bobi\'nski presents a different way of determining the
Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of
Geigle.
