Characters actually start with SEVEN aspects. The five that you mentioned, gained from the background mini-stories, plus a High Concept aspect and a Trouble aspect representing the core of the character's identity and the thing that makes their life most interesting in the proverbial sense, respectively.
All other aspects that a character might gain during the course of play are generally temporary, placed by maneuvers/declarations/assessments, consequential contests, and Consequences generated by attacks.
Barring GM fiat (meaning that, regardless of what the rules might say, the GM can say otherwise up to the point where his players object enough that they all up and leave), characters do not gain any more permanent aspects than the 7 that they started with.
Characters grow stronger / more experienced when they reach Milestones. Depending on the importance of the Milestone, this may allow them to simply switch around a few things on their sheet if they're not happy with them (or as representing a change in priorities/etc in the character's life), or it might give the character another skill point to spend (keeping the 'skill pyramid' in mind) as well as allowing them to move some things around, or it might do both of those as well as giving the character another point of Refresh to spend (or save).
More detail on Milestones can be found on page 88.