We explore the existence of tightly confined gap modes in structures consisting of two infinitely long graphene ribbons vertically offset by a gap. By investigating carefully such a sandwich geometry we find that the gap modes originate from a strong hybridization that gives rise to improved waveguide performance while modifying the guiding behaviour compared to a single ribbon. Our work particularly focuses on the physical origin and description of these plasmon modes, studying the critical parameters of width, gap and operation wavelength. This allows different regimes, coupling mechanisms and mode families to be recognized. Importantly we show that the gap modes also exist when a single graphene sheet is placed on top of a metal or a doped semiconductor—a geometry that is readily achievable experimentally. As an example we report on an unprecedented level of confinement of a terahertz wave of nearly five orders of magnitude when a graphene ribbon is placed on top of a highly doped silicon substrate. Because of their remarkable field distributions and extreme confinement, the families of modes presented here could be the building blocks for both graphene-based integrated optics and ultrasensitive sensing modalities.