High-entropy ceramics (HECs) have quickly gained attention since 2015. To date, nearly all work has focused on five-component, equimolar compositions. This perspective article briefly reviews different families of HECs and selected properties. Following a couple of our most recent studies, we propose a step forward to expand HECs to compositionally complex ceramics (CCCs) to include medium-entropy and non-equimolar compositions. Using defective fluorite and ordered pyrochlore oxides as two primary examples, we further consider the complexities of aliovalent cations and anion vacancies as well as ordered structures with two cation sublattices. Better thermally insulating yet stiff CCCs have been found in non-equimolar compositions with optimal amounts of oxygen vacancies and in ordered pyrochlores with substantial size disorder. It is demonstrated that medium-entropy ceramics can prevail over their high-entropy counterparts. The diversifying classes of CCCs provide even more possibilities than HECs to tailor the composition, defects, disorder/order, and, consequently, various properties.