Caching is popular technique in content delivery networks that allows for reductions in transmission rates from the content-hosting server to the end users. Coded caching is a generalization of conventional caching that considers the possibility of coding in the caches and transmitting coded signals from the server. Prior results in this area demonstrate that huge reductions in transmission rates are possible and this makes coded caching an attractive option for the next generation of content-delivery networks. However, these results require that each file hosted in the server be partitioned into a large number (i.e., the subpacketization level) of non- overlapping subfiles. From a practical perspective, this is problematic as it means that prior schemes are only applicable when the size of the files is extremely large. In this work, we propose a novel coded caching scheme that enjoys a significantly lower subpacketization level than prior schemes, while only suffering a marginal increase in the transmission rate. In particular, for a fixed cache size, the scaling with the number of users is such that the increase in transmission rate is negligible, but the decrease in subpacketization level is exponential.