We call edgelet computing the current convergence between Opportunistic Network (OppNet) and Trusted Execution Environment (TEE) at the very edge of the network. We believe that this convergence bears the seeds of a novel and important class of applications leveraging fully decentralized and highly secure computations among data scattered on multiple personal devices. This paper introduces the Edgelet computing paradigm, defines properties that guarantee the safety, liveness and security of executions in this unusual context and proposes alternative strategies satisfying these properties. Preliminary performance evaluations and an ongoing real-case study highlights the practicality of the approach. Finally, the paper draws future research challenges for the database and distributed system community.