Quantum batteries provide a setting to investigate whether quantum effects can enhance the rate of energy storage beyond classical limits. While superextensive charging power has been reported in a few models, the fundamental physical requirements for achieving such scaling remain unclear. Here we identify the growth with system size of the $g$-extensiveness of either the battery or charging Hamiltonian as a necessary condition for achieving superextensive charging power in direct-charging protocols. We prove that bounded $g$-extensiveness forbids superextensive power, irrespective of interaction order. This criterion also explains why many previously studied protocols fail to exhibit superextensive behavior despite involving interactions with large participation numbers. Crucially, we demonstrate that global operations are not necessary: superextensive power can be achieved using Hamiltonians of fixed interaction order that remain energy extensive, provided $g$-extensiveness grows with system size. We illustrate this mechanism using a modified central-spin model.

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• Quantum correlations in quantum information
• Quantum foundations
• Quantum protocols
• Quantum thermodynamics