We use off-axis electron holography to measure the electrostatic charge density distributions on graphene-based nanogap devices that have thicknesses of between 1 and 10 monolayers and separations of between 8 and 58 nm with a precision of better than a single unit charge. Our experimental measurements, which are compared with finite element simulations, show that wider graphene tips, which have thicknesses of a single monolayer at their ends, exhibit charge accumulation along their edges. The results are relevant for both fundamental research on graphene electrostatics and applications of graphene nanogaps to single nucleotide detection in DNA sequencing, single molecule electronics, plasmonic antennae, and cold field emission sources.