Actually, it looks like any party can also object to the recount itself and present evidence (such as a lead of a wide margin) and from what it appears, each election unit can make a finding of fact based upon that evidence. So, for example, Hillary's campaign could object (there's no way that there's a million votes sitting around which would give Jill Stein the victory and this is just a waste of time) and each body could make a determination (yes, there's no way that any number of recounts would make Jill Stein the victor and this is just a huge waste of time.)
What appears to be absent would be a state level decision of that type; the way it is written all objections must be made with the body doing each recount.
I’m going by the letter sent from the Elections Supervisor. He says if Stein pays, the recount is on:
http://www.freerepublic.com/focus/news/3498409/posts?page=20#20