The ongoing Big Data explosion has created a demand for efficient and scalable algorithms for similarity search. Most recent work has focused on \textit{approximate} $k$-NN search, and while this may be sufficient for some applications, \textit{exact} $k$-NN search would be ideal for many applications. We present CAKES, a set of three novel, exact algorithms for $k$-NN search. CAKES's algorithms are generic over \textit{any} distance function, and they \textit{do not} scale with the cardinality or embedding dimension of the dataset, but rather with its metric entropy and fractal dimension. We test these claims on datasets from the ANN-Benchmarks suite under commonly-used distance functions, as well as on a genomic dataset with Levenshtein distance and a radio-frequency dataset with Dynamic Time Warping distance. We demonstrate that CAKES exhibits near-constant scaling with cardinality on data conforming to the manifold hypothesis, and has perfect recall on data in \textit{metric} spaces. We also demonstrate that CAKES exhibits significantly higher recall than state-of-the-art $k$-NN search algorithms when the distance function is not a metric. Additionally, we show that indexing and tuning time for CAKES is an order of magnitude, or more, faster than state-of-the-art approaches. We conclude that CAKES is a highly efficient and scalable algorithm for exact $k$-NN search on Big Data. We provide a Rust implementation of CAKES under an MIT license at https://github.com/URI-ABD/clam