Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (by his letter) entirely out of sight: Declared as "national treasure", they seem to be in principle accessible (+ Thanks to Jonathan Chiche who points - see his comment below - that it is not so clear if that idea was made a reality by now): http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-maths_830399
On p. 185 - 186 of the 3rd volume of Winfried Scharlau's Grothendieck biography, a handwritten text from 1986 by Grothendieck on foundations of topology, different from the concepts of topoi or tame topology, is shortly described. Scharlau doubts if it could be turned into a readable text, but perhaps someone knows the texts and has ideas about it?
Edit: Acc. to Winfried Scharlau's book, Grothendieck described his work in a letter to Jun-Ichi Yamashita as: "some altogether different foundations of 'topology', starting with the 'geometrical objects' or 'figures', rather than starting with a set of 'points' and some kind of notion of 'limit' or (equivalently) 'neighbourhoods'. Like the language of topoi (and unlike 'tame topology'), it is a kind of topology 'without points' - a direct approach to 'shape'. ... appropriate for dealing with finite spaces... the mathematics of infinity are just a way of approximating an understanding of finite agregates, whose structures seem too elusive or too hopelessly intricate for a more direct understanding (at least it has been until now)." Scharlau gives a copy of one page of the manuscript (at p. 188) and obviously has a copy of the complete text and remarks (on p. 199) that Grothendieck wrote a in 1983 letter about that theme to Z. Mebkhout.
Edit: In the meantime I could read a letter by Grothendieck about that, a summary: He started thinking from time to time about that ca. in the mid-1970's, the motivation was roughly that dissatisfaction with the usual topology which he expressed in the Esquisse, and looking at stratifications of moduli-"spaces" is his new starting point. Maybe, but not expressed in the letter or the Esquisse, the ubiquity of moduli problems in algebraic geometry (e.g. expressed in the beginning of Lafforgue's text ) is an other motivation. He describes his guiding ideas on new foundations of topology as more complicated than the guiding ideas behind the new foundations of algebraic geometry of EGA, SGA. A main test of his concepts now would be a "Dévissage"-theorem on "startified obstructions"(?) in terms of equivalences of categories. He has a precise heuristic formulation of that which helped him to find a "dévissage" corresponding to Teichmueller groups (probably what now is called "Grothendieck-Teichmueller group"?) which are related to stratifications "at infinity" of Deligne-Mumford moduli stacks.