In view of the many comments, I will make a (I hope correct) summary of these comments in CW mode; everybody please feel free to edit:
If one starts with a completely 'blank sheet of paper' it seems that one can do almost nothing with a straightedge alone.
However, as mentioned by François Brunault given certain 'initial constellations' one can construct some additional interesting points using a staightedege alone (see here (in French)).
Daniel Briggs suggested to 'add' just one circle with known center (the unit circle). If one does this, then by the Poncelet-Steiner theorem (mentioned by François Brunault) one can already construct everything one can construct with straightedge and compass.
L. Spice mentioned that by the Mohr-Mascheroni theorem the converse situation (only a compass no straightedge) allows also to construct everything one can construct with straightedge and compass.
The book Leçons sur les constructions géométriques by Lebesgue is entirely devoted to the question of geometric constructions using various instruments. The table of contents of this book (in French, again) is available here.
