Many convex optimization problems can be represented through conic extended formulations with auxiliary variables and constraints using only the small number of standard cones recognized by advanced conic solvers such as MOSEK 9. Such extended formulations are often significantly larger and more complex than equivalent conic natural formulations, which can use a much broader class of exotic cones. We define an exotic cone as a proper cone for which we can implement efficient logarithmically homogeneous self-concordant barrier oracles for the cone or its dual. Our goal is to establish whether a generic conic interior point method supporting natural formulations can outperform an advanced conic solver specialized for standard cones. We introduce Hypatia, a highly-configurable open-source conic primal-dual interior point solver with a generic interface for exotic cones. Hypatia is written in Julia and accessible through JuMP, and currently implements several dozen useful predefined cones. We define a subset of these cones, including some that have not been implemented before, and we propose several new efficient logarithmically homogeneous self-concordant barriers. We also describe and analyze techniques for constructing extended formulations for exotic conic constraints. For optimization problems from a variety of applications, we introduce natural formulations using our exotic cones, and we show that the natural formulations have much smaller dimensions and often lower barrier parameters than the equivalent extended formulations. Our computational experiments demonstrate the potential advantages, especially in terms of solve time and memory usage, of solving natural formulations with Hypatia compared to solving extended formulations with Hypatia or MOSEK.