The structural network of the human brain has a rich topology which many have sought to characterise using standard network science measures and concepts. However, this characterisation remains incomplete and the non-obvious features of this topology have largely confounded attempts towards comprehensive constructive modelling. This calls for new perspectives. Hierarchical complexity is an emerging paradigm of complex network topology based on the observation that complex systems are composed of hierarchies within which the roles of hierarchically equivalent nodes display highly variable connectivity patterns. Here we test the hierarchical complexity of the human structural connectomes of a group of seventy-nine healthy adults. Binary connectomes are found to be more hierarchically complex than three benchmark random network models. This provides a new key description of brain structure, revealing a rich diversity of connectivity patterns within hierarchically equivalent nodes. Dividing the connectomes into four tiers based on degree magnitudes indicates that the most complex nodes are neither those with the highest nor lowest degrees but are instead found in the middle tiers. Spatial mapping of the brain regions in each hierarchical tier reveals consistency with the current anatomical, functional and neuropsychological knowledge of the human brain. The most complex tier (Tier 3) involves regions believed to bridge high-order cognitive (Tier 1) and low-order sensorimotor processing (Tier 2). We then show that such diversity of connectivity patterns aligns with the diversity of functional roles played out across the brain, demonstrating that hierarchical complexity can characterise functional diversity strictly from the network topology.