There's no field with one element in the literal sense, but there are constructions that work over different fields `F_q` and which sometimes make sense when `q=1`. Examples would include representation theory of `GL_n(F_q)` which, if I'm correct, becomes the representation theory of `S_n` under that limit. 


In particular, indeed, vector spaces — the objects on which `GL_n(F_q)` act — should become sets, the objects on which `S_n` acts.

Though I'm not an expert on `F_1`, I've encountered the viewpoint you're referring to. It's been called `absolute motives` in the works I read, I think by Manin, Kontsevich, and some IHES people. Indeed, the think that people talk about is the mental picture of every scheme being kind a scheme defined over `F_1`. (The search for `kontsevich absolute motives` brought me up the article that I found a while ago, **[math/0702206][1]**)

This is also somehow related to noncommutative geometry of Connes, in fact I think `F_1` is one of his favorite ideas (or is it from some other part of French math circle?) 

Also, here's a [blog post about `F_1`][2] with link to the introductory paper. In fact there was a whole blog about `F_un`, which strangely disappeared (well, at least the articles I wanted to read should be still somewhere cached at my other computer).


  [1]: http://arxiv.org/abs/math/0702206
  [2]: http://noncommutativegeometry.blogspot.com/2008/06/fun-day-two.html