We present a novel approach to perform agglomeration of mesh elements based on R-tree spatial indices. Applied to an arbitrary fine mesh, the R-tree offers a natural and efficient mesh agglomeration strategy. The process is fully automated, robust, and dimension-independent. Moreover, t is guaranteed to produce a balanced and nested hierarchy of meshes. R-tree mesh agglomerates are general polygons and polyhedra. However, their shape is tightly close to the respective axis aligned bounding boxes, even if the underlying fine mesh is unstructured. Hence the procedure can be seen as a way to generalise standard box-like meshes on which polygonal and polyhedral (polytopic) FEMs can be implemented efficiently. We showcase their use with the polytopic discontinuous Galerkin method which we prove to be stable in the setting of general agglomerated meshes. Further, we exploit the full hierarchy of nested agglomerates and fast query capabilities of the R-tree approach within efficient geometric multigrid methods to be applied to those cases where a hierarchy of grids is not present at construction time.