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.
There's more to say about F_1 from the point of view of absolute motives, but I'm not an expert (one think that people talk about is the mental picture of absolutely every scheme being kind of a scheme defined over F_1, search for kontsevich absolute motives). This is also related to 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 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).