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The electrochemical (EC) behavior of mechanically exfoliated graphene and highly oriented pyrolytic graphite (HOPG) is studied at high spatial resolution in aqueous solutions using Ru(NH3)63+/2+as a redox probe whose standard potential sits close to the intrinsic Fermi level of graphene and graphite. When scanning electrochemical cell microscopy (SECCM) data are coupled with that from complementary techniques (AFM, micro-Raman) applied to the same sample area, different time-dependent EC activity between the basal planes and step edges is revealed. In contrast, other redox couples (ferrocene derivatives) whose potential is further removed from the intrinsic Fermi level of graphene and graphite show uniform and high activity (close to diffusion-control). Macroscopic voltammetric measurements in different environments reveal that the time-dependent behavior after HOPG cleavage, peculiar to Ru(NH3)63+/2+, is not associated particularly with any surface contaminants but is reasonably attributed to the spontaneous delamination of the HOPG with time to create partially coupled graphene layers, further supported by conductive AFM measurements. This process has a major impact on the density of states of graphene and graphite edges, particularly at the intrinsic Fermi level to which Ru(NH3)63+/2+is most sensitive. Through the use of an improved voltammetric mode of SECCM, we produce movies of potential-resolved and spatially resolved HOPG activity, revealing how enhanced activity at step edges is a subtle effect for Ru(NH3)63+/2+. These latter studies allow us to propose a microscopic model to interpret the EC response of graphene (basal plane and edges) and aged HOPG considering the nontrivial electronic band structure.