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For tabletop object rearrangement problems with overhand grasps, storage space which may be inside or outside the tabletop workspace, or running buffers, can temporarily hold objects which greatly facilitates the resolution of a given rearrangement task. This brings forth the natural question of how many running buffers are required so that certain classes of tabletop rearrangement problems are feasible. In this work, we examine the problem for both the labeled (where each object has a specific goal pose) and the unlabeled (where goal poses of objects are interchangeable) settings. On the structural side, we observe that finding the minimum number of running buffers (MRB) can be carried out on a dependency graph abstracted from a problem instance, and show that computing MRB on dependency graphs is NP-hard. We then prove that under both labeled and unlabeled settings, even for uniform cylindrical objects, the number of required running buffers may grow unbounded as the number of objects to be rearranged increases; we further show that the bound for the unlabeled case is tight. On the algorithmic side, we develop highly effective, exact algorithms for finding MRB for both labeled and unlabeled tabletop rearrangement problems, scalable to over a hundred objects under very high object density. More importantly, our algorithms also compute a sequence witnessing the computed MRB that can be used for solving object rearrangement tasks. Employing these algorithms, empirical evaluations reveal that random labeled and unlabeled instances, which more closely mimics real-world setups, generally have fairly small MRBs.