The Heliospheric Imagers on board NASAs twin STEREO spacecraft show that coronal mass ejections (CMEs) can be visually complex structures. To explore this complexity, we created a web-based citizen science project in collaboration with the UK Science Museum, in which participants were shown pairs of differenced CME images, taken by the inner cameras of the Heliospheric Imagers (HI-1) on board the twin NASA STEREO spacecraft between 2008 and 2016. Participants were asked to decide which image in each pair appeared the most complicated. 4,028 volunteers conducted 246,692 comparisons of 20,190 image pairs, with each pair being classified by 12 independent users. A Bradley-Terry model was then applied to these data to rank the CMEs by their visual complexity. This complexity ranking revealed that the annual average visual complexity values follow the solar activity cycle, with a higher level of complexity being observed at the peak of the cycle, and the average complexity of CMEs observed by HI1-A was significantly higher than the complexity of CMEs observed by HI1-B. Visual complexity was found to be associated with CME size and brightness, but the differences between HI1-A and HI1-B images suggest that complexity may be influenced by the scale-sizes of structure in the CMEs. Whilst it might not be surprising that the complexity observed in these CME images follows the trend observed in sunspots and the solar cycle; these results demonstrate that there is a quantifiable change in the structure of CMEs seen in the inner heliosphere.