This paper investigates a cellular edge caching design under an extremely large number of small base stations (SBSs) and users. In this ultra-dense edge caching network (UDCN), SBS-user distances shrink, and each user can request a cached content from multiple SBSs. Unfortunately, existing caching controls’ complexity increases with the number of SBSs, making them inapplicable for solving the caching trade-off, i.e., maximizing local caching gain while minimizing the replicated content caching. Furthermore, spatial dynamics of interference is no longer negligible in UDCNs due to the surge in interference amount and variance. In addition, the caching control should consider temporal dynamics of user demand. Only a short-term misprediction significantly exacerbates the caching performance because of the populous SBSs and users. To overcome such difficulties, we propose a novel caching algorithm leveraging the frameworks of mean-field game and stochastic geometry. These enable our algorithm to computationally become independent of the number of SBSs and users, while incorporating spatial interference dynamics as well as temporal dynamics of content popularity and storage space. Numerical evaluation validates that the proposed algorithm reduces not only long run average cost by at least 24% but also the replicated content amount averagely 56% compared to a popularity-based algorithm.