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Edge computing has enabled a large set of emerging edge applications by exploiting data proximity and offloading computation-intensive workloads to nearby edge servers. However, supporting edge application users at scale poses challenges due to limited point-of-presence edge sites and constrained elasticity. In this paper, we introduce a densely-distributed edge resource model that leverages capacity-constrained volunteer edge nodes to support elastic computation offloading. Our model also enables the use of geo-distributed edge nodes to further support elasticity. Collectively, these features raise the issue of edge selection. We present a distributed edge selection approach that relies on client-centric views of available edge nodes to optimize average end-to-end latency, with considerations of system heterogeneity, resource contention and node churn. Elasticity is achieved by fine-grained performance probing, dynamic load balancing, and proactive multi-edge node connections per client. Evaluations are conducted in both real-world volunteer environments and emulated platforms to show how a common edge application, namely AR-based cognitive assistance, can benefit from our approach and deliver low-latency responses to distributed users at scale.