My favorite one is the ancient Chinese classification of animals from the
Encyclopaedia of Benevolent Knowledge:
Unfortunately, the primary source did not survive
A hierarchic process of producing taxonomies was formalized by 3rd century AD logician Porphyry in his presentation of Aristotle's classification of categories, and became popular during middle ages in a diagrammatic form of "Porphyrian tree". One starts with the highest category in a field, "summum genus", and proceeds in successive steps by using "...
Linnean taxonomy was very much the progression and refinement of earlier taxonomic thoughts. This answer draws heavily from "Linnaeus as an Intermediary between Ancient and Modern Zoölogy; His Views on the Class Mammalia" by W. K. Gregory (1908).
Gregory notes several major figures in the history of taxonomic nomenclature before Linnaeus: