We report a systematic study of the total contact resistance present at the interface between a metal (Ti) and graphene layers of different, known thickness. By comparing devices fabricated on many different graphene flakes we demonstrate that the contact resistance consists of a gate independent and a gate dependent part. We show that quantitatively the gate independent part of the contact resistance is the same for single-, bi-, and tri-layer graphene. We argue that this is the result of charge transfer from the metal, causing the Fermi level in the graphene region under the contacts to shift far away from the charge neutrality point.