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The growing demand for well-modeled ontologies in diverse application areas increases the need for intuitive interaction techniques that support human domain experts in ontology modeling and enrichment tasks, such that quality expectations are met. Beyond the correctness of the specified information, the quality of an ontology depends on its (relative) completeness, i.e., whether the ontology contains all the necessary information to draw expected inferences. On an abstract level, the Ontology Enrichment problem consists of identifying and filling the gap between information that can be logically inferred from the ontology and the information expected to be inferable by the user. To this end, numerous approaches have been described in the literature, providing methodologies from the fields of Formal Semantics and Automated Reasoning targeted at eliciting knowledge from human domain experts. These approaches vary greatly in many aspects and their applicability typically depends on the specifics of the concrete modeling scenario at hand. Toward a better understanding of the landscape of methodological possibilities, this position paper proposes a framework consisting of multiple performance dimensions along which existing and future approaches to interactive ontology enrichment can be characterized. We apply our categorization scheme to a selection of methodologies from the literature. In light of this comparison, we address the limitations of the methods and propose directions for future work.