http://mathoverflow.net/questions/103492/grothendieck-topologies-versus-pretopologies
http://mathoverflow.net/questions/80074/commuting-grothendieck-topologies
http://mathoverflow.net/questions/34717/analytic-tools-in-algebraic-geometry
de Jong on proper hypercoverings: http://math.columbia.edu/~dejong/wordpress/?p=2190 and on cocontinuous functors: http://math.columbia.edu/~dejong/wordpress/?p=2149
Several notions of Gabber (alteration topology, etc) are explained in the book on arxiv by Illusie, Laszlo and Orgogozo on the works of Gabber.
Suslin and Voevodsky: Singular homology of abstract algebraic varieties. They also have a few pages on the qfh and h topologies and their sheaves.
Voevodsky: Unstable motivic homotopy categories in Nisnevich and cdg topologies.
Milne seems to think that a suff fine top computes the true cohom, and a natural question is how coarse can we go and still get the same coh. See e.g. Levine MM section 5.2.
A site of Zink, maybe same as above. Link
Bondarko claims in an MO question, that in the setting of a closed embedding from Z to X: “we have a cartesian square of sites: Xet, Xnis, Zet, Znis, and there are also sites over the Zariski points of X in this picture”.
nLab page on Grothendieck topology III