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The high current-carrying capacity of graphene is essential for its use as an interconnect in electronic and spintronic circuits. At the same time, knowing the breakdown limits and mechanism under high fields can enable new device design strategies. In this work, we push the current carrying capacity of the scalable form of chemical vapor deposited (CVD) graphene employing a high-thermal conducting single crystalline diamond substrate. Our experiments on CVD graphene reveal extremely high current densities > 10⁹ A/cm² in graphene on the diamond with both ohmic (low-resistive) and tunneling tunnel (high-resistive) contacts. Measurements on ferromagnetic (TiO_x/Co) and metallic (Ti/Au) contacts demonstrate current densities of ∼1.16×10⁹ A/cm² and ∼1.7×10⁹ A/cm², respectively. The tunnel (high-resistive) contacts exhibit a shunting of graphene under high currents via the bottom graphitized diamond, resulting in dielectric breakdown and via alternative conducting paths. Electrical measurements show a distinct threshold for conducting paths of graphitized diamond, in tune accordance with Middleton-Wingreen's theory. Our results of high current densities achieved in CVD graphene, with distinct dependence on ohmic and tunneling, contact resistance, and the observed breakdown mechanism, provide new insights for enabling high-current all carbon circuits.