Real complex systems are not rigidly structured; no clear rules or blueprints exist for their construction. Yet, amidst their apparent randomness, complex structural properties appear to universally emerge. We propose that an important class of complex systems can be modelled as a construction of potentially infinitely many levels of organization all following the same universal growth principle known as preferential attachment. We give examples of such hierarchy in real systems, for instance in the pyramid of production entities of the movie industry. More importantly, we show how real complex networks can be interpreted as a projection of our model, from which their scale independence, their clustering or modularity, their hierarchy, their fractality and their navigability naturally emerge. Our results suggest that complex networks, viewed as growing systems, can be quite simple, and that the apparent complexity of their structure is largely a reflection of the hierarchical nature of our world.