Focused-electron-beam-induced deposition, or FEBID, enables the fabrication of patterns with sub-10 nm resolution. The initial stages of metal deposition by FEBID are still not fundamentally well understood. For these investigations, graphene, a one-atom-thick sheet of carbon atoms in a hexagonal lattice, is ideal as the substrate for FEBID writing. In this paper, we have used exfoliated few-layer graphene as a support to study the early growth phase of focused-electron-beam-induced deposition and to write patterns with dimensions between 0.6 and 5 nm. The results obtained here are compared to the deposition behavior on amorphous materials. Prior to the deposition experiment, the few-layer graphene was cleaned. Typically, it is observed in electron microscope images that areas of microscopically clean graphene are surrounded by areas with amorphous material. We present a method to remove the amorphous material in order to obtain large areas of microscopically clean graphene flakes. After cleaning, W(CO)6 was used as the precursor to study the early growth phase of FEBID deposits. It was observed that preferential adsorption of the precursor molecules on step edges and adsorbates plays a key role in the deposition on cleaned few-layer graphene.