Caching has the potential to be of significant benefit for accessing large language models (LLMs) due to their high latencies which typically range from a small number of seconds to well over a minute. Furthermore, many LLMs charge money for queries; caching thus has a clear monetary benefit. This paper presents a new caching system for improving user experiences with LLMs. In addition to reducing both latencies and monetary costs for accessing LLMs, our system also provides important features that go beyond the performance benefits typically associated with caches. A key feature we provide is generative caching, wherein multiple cached responses can be synthesized to provide answers to queries which have never been seen before. Our generative caches function as repositories of valuable information which can be mined and analyzed. We also improve upon past semantic caching techniques by tailoring the caching algorithms to optimally balance cost and latency reduction with the quality of responses provided. Performance tests indicate that our caches are considerably faster than GPTcache.