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For nearly six decades, the central open question in the study of hash tables has been to determine the optimal achievable tradeoff curve between time and space. State-of-the-art hash tables offer the following guarantee: If keys/values are Θ(logn) bits each, then it is possible to achieve constant-time insertions/deletions/queries while wasting only O(loglogn) bits of space per key when compared to the information-theoretic optimum. Even prior to this bound being achieved, the target of O(log log n) wasted bits per key was known to be a natural end goal, and was proven to be optimal for a number of closely related problems (e.g., stable hashing, dynamic retrieval, and dynamically-resized filters). This paper shows that O(log log n) wasted bits per key is not the end of the line for hashing. In fact, for any k ∈ [log∗ n], it is possible to achieve O(k)-time insertions/deletions, O(1)-time queries, and O(log(k) n) = Ologlog···logn 􏰟 􏰞􏰝 􏰠 k wasted bits per key (all with high probability in n). This means that, each time we increase inser- tion/deletion time by an additive constant, we reduce the wasted bits per key exponentially. We further show that this tradeoff curve is the best achievable by any of a large class of hash tables, including any hash table designed using the current framework for making constant-time hash tables succinct. Our results hold not just for fixed-capacity hash tables, but also for hash tables that are dynamically resized (this is a fundamental departure from what is possible for filters); and for hash tables that store very large keys/values, each of which can be up to no(1) bits (this breaks with the conventional wisdom that larger keys/values should lead to more wasted bits per key). For very small keys/values, we are able to tighten our bounds to o(1) wasted bits per key, even when k = O(1). Building on this, we obtain a constant-time dynamic filter that uses n􏰕logε−1􏰖+nloge+o(n) bits of space for a wide choice of