Modern Web APIs allow developers to provide extensively customized experiences for website visitors, but the richness of the device information they provide also make them vulnerable to being abused to construct browser fingerprints, device-specific identifiers that enable covert tracking of users even when cookies are disabled. Previous research has established entropy, a measure of information, as the key metric for quantifying fingerprinting risk. However, earlier studies had two major limitations. First, their entropy estimates were based on either a single website or a very small sample of devices. Second, they did not adequately consider correlations among different Web APIs, potentially grossly overestimating their fingerprinting risk. We provide the first study of browser fingerprinting which addresses the limitations of prior work. Our study is based on actual visited pages and Web APIs reported by tens of millions of real Chrome browsers in-the-wild. We accounted for the dependencies and correlations among Web APIs, which is crucial for obtaining more realistic entropy estimates. We also developed a novel experimental design that accurately and efficiently estimates entropy while never observing too much information from any single user. Our results provide an understanding of the distribution of entropy for different website categories, confirm the utility of entropy as a fingerprinting proxy, and offer a method for evaluating browser enhancements which are intended to mitigate fingerprinting.