If you had two "perfectly" flat surfaces of the same material? Let's say you had 2 nano-engineered surfaces of diamond which were as 'flat' as possible (of course considering the radii of each carbon atom in the lattice)... would there be any friction between these 2 flat diamond surfaces when rubbed together? 
My thinking is that there would be very little to no friction due to the electron repulsion and there being no imperfections in the surfaces.
 A: This is a footnote to Chris' answer (+1 BTW :-)
The microsocopic source of friction is exactly the cold welding mentioned by Chris and DJBunk. When you touch two rough surfaces together their real area of contact will be much less than the total area because only the high spots touch. At these high spots the surfaces adhere due to the same forces that act in the bulk (give or take a bit of surface contamination). Friction exists because to slide the surfaces you have to fracture them along the contact points and this takes energy.
The usual (approximate) formula for friction, $F_f = \mu F_n$, arises because the real area of contact is roughly proportional to load. As you increase the load you deform the points where the surfaces touch and therefore increase the real area of contact and adhesion.
Your example of a perfectly smooth surface is simply the limit of the real area of contact becoming equal to the total area, which in principle would happen at extremely high loads. The result (again ignoring surface contamination) would be that the surfaces fuse with at worst a grain boundary between them.
A: Quite likely, the two materials would stick together and form a seamless bond. If you have two identical crystal lattices, and each one is bond-deficient at the surface, it will be energetically favorable for the surfaces to bond.
Moreover, by making the surfaces as flat as possible, you have made it likely that large-scale alignment will occur. Think of a few atoms - A1, A2, maybe A3 - in crystal A being in locations that naturally extend the lattice of crystal B. Then the translational symmetry of the lattices means all the atoms on the surface of A are at lattice points for B.
The crystals will snap together, with little or no evidence that they were ever separate. This is the basis for cold welding, as mentioned in the comments.
