Embedding a clause inside another (``the girl [who likes cars [that run fast]] has arrived″) is a fundamental resource that has been argued to be a key driver of linguistic expressiveness. As such, it plays a central role in fundamental debates on what makes human language unique, and how they might have evolved. Empirical evidence on the prevalence and the limits of embeddings has however been based on either laboratory setups or corpus data of relatively limited size. We introduce here a collection of large, dependency-parsed written corpora in 17 languages, that allow us, for the first time, to capture clausal embedding through dependency graphs and assess their distribution. Our results indicate that there is no evidence for hard constraints on embedding depth: the tail of depth distributions is heavy. Moreover, although deeply embedded clauses tend to be shorter, suggesting processing load issues, complex sentences with many embeddings do not display a bias towards less deep embeddings. Taken together, the results suggest that deep embeddings are not disfavoured in written language. More generally, our study illustrates how resources and methods from latest-generation big-data NLP can provide new perspectives on fundamental questions in theoretical linguistics.