Dense retrieval, which encodes queries and documents into a single dense vector, has become the dominant neural retrieval approach due to its simplicity and compatibility with fast approximate nearest neighbor algorithms. As the tasks dense retrieval performs grow in complexity, the fundamental limitations of the underlying data structure and similarity metric---namely vectors and inner-products---become more apparent. Recent work has shown theoretical limitations inherent to single vectors and inner-products, which are generally tied to the embedding dimension. Given the importance of embedding dimension for retrieval capacity, understanding how dense retrieval performance changes as embedding dimension is scaled is fundamental to building next-generation retrieval models that balance effectiveness and efficiency. In this work, we conduct a comprehensive analysis of the relationship between embedding dimension and retrieval performance. Our experiments include a range of model sizes from two model families to construct a detailed picture of embedding scaling behavior. We find that the scaling behavior fits a power law, allowing us to derive scaling laws for performance given only the embedding dimension, as well as a joint law accounting for embedding dimension and model size. Our analysis shows that for evaluation tasks aligned with the training task, performance continues to improve as embedding size increases, though with diminishing returns. For evaluation data that are less aligned with the training task, we find that performance is less predictable, with performance degrading with larger embedding dimensions for certain tasks. We hope our work provides additional insight into the limitations of embeddings and their behavior as well as offers a practical guide for selecting model and embedding dimension to achieve optimal performance with reduced storage and compute costs.