Long-term memory is content addressable, in that partial cues are sufficient to drive recognition or recall of complete objects and events. Several content addressable memory (CAM) architectures have been proposed to model long-term memory, including the Hopfield network, several variants of the Hopfield network, and overparametrized autoencoders. However, all of these architectures exhibit a memory cliff, beyond which adding a single pattern leads to catastrophic loss of all patterns. Here we propose a novel and biologically motivated memory architecture, Memory through Scaffolded Heteroassociation (MESH), that generates a CAM continuum: storage of information-dense patterns up to a critical capacity results in complete recovery of all patterns and storage of a larger number of patterns results in partial reconstruction of the stored patterns. This partial reconstruction continues up to an exponentially large number of patterns resulting in correct recognition of each of the stored patterns. Inspired by the entorhinal-hippocampal circuit, MESH contains a bipartite attractor network that stores a large dictionary of well-separated fixed points that serve as a pre-defined ``memory scaffold''. Arbitrary dense patterns are then stored by associating them to the pre-defined scaffold states. This novel combination of predefined attractor states along with heteroassociative learning that hooks patterns on to scaffolding states results in a biologically plausible CAM continuum, that approaches the theoretical upper-bound on information storage in neural networks. We believe that this is the first model of a content-addressable memory that automatically trades off pattern number and pattern richness; it makes the testable prediction that biological memory systems may exploit pre-existing scaffolds to acquire new memories, potentially consistent with the preplay of hippocampal sequences before they are used for representing new environments.