Current Approximate Query Processing (AQP) engines are far from silver-bullet solutions, as they adopt several static design decisions that target specific workloads and deployment scenarios. Offline AQP engines target deployments with large storage budget, and offer substantial performance improvement for predictable workloads, but fail when new query types appear, i.e., due to shifting user interests. To the other extreme, online AQP engines assume that query workloads are unpredictable, and therefore build all samples at query time, without reusing samples (or parts of them) across queries. Clearly, both extremes miss out on different opportunities for optimizing performance and cost. In this paper, we present Taster, a self-tuning, elastic, online AQP engine that synergistically combines the benefits of online and offline AQP. Taster performs online approximation by injecting synopses (samples and sketches) into the query plan, while at the same time it strategically materializes and reuses synopses across queries, and continuously adapts them to changes in the workload and to the available storage resources. Our experimental evaluation shows that Taster adapts to shifting workload and to varying storage budgets, and always matches or significantly outperforms the state-of-the-art performing AQP approaches (online or offline).