Class-Agnostic object Counting (CAC) involves counting instances of objects from arbitrary classes within an image. Due to its practical importance, CAC has received increasing attention in recent years. Most existing methods assume a single object class per image, rely on extensive training of large deep learning models and address the problem by incorporating additional information, such as visual exemplars or text prompts. In this paper, we present OCCAM, the first training-free approach to CAC that operates without the need of any supplementary information. Moreover, our approach addresses the multi-class variant of the problem, as it is capable of counting the object instances in each and every class among arbitrary object classes within an image. We leverage Segment Anything Model 2 (SAM2), a foundation model, and a custom threshold-based variant of the First Integer Neighbor Clustering Hierarchy (FINCH) algorithm to achieve competitive performance on widely used benchmark datasets, FSC-147 and CARPK. We propose a synthetic multi-class dataset and F1 score as a more suitable evaluation metric. The code for our method and the proposed synthetic dataset will be made publicly available at https://mikespanak.github.io/OCCAM_counter.