I have a Hardened Scales and Setessan Oathsworn with no counters on the battlefield. I cast Solidarity of Heroes on the Oathsworn, it gets three counters from it's heroic trigger which happens while the Solidarity is still on the stack. When the Solidarity resolves will it have 6 counters total or 7? Does doubling counters translate to "add as many counters as it has" or is it some special rule that changes the count without "adding" (thus not triggering the Scales).
When the card says to "double the number of +1/+1 counters" on a creature, it means that you should place on to that creature as many +1/+1 counters as are currently on it. This means that Hardened Scales applies, so Setessan Oathsworn will end up with 7 +1/+1 counters on it.
Unfortunately, I can't find a rules reference to back this up, but the basic idea is that after Solidarity of Heroes resolves, there are more counters on the creature than before it resolved, and the only way that could have happened (from the point of view of the game rules) is if you put counters on it. There is no other applicable action that has that result.
The relevant rule seems to be:
121.6. If a spell or ability refers to a counter being "placed" on a permanent, it means putting a counter on that permanent while it's on the battlefield, or that permanent entering the battlefield with a counter on it as the result of an effect (see rule 614.1c).
It is not entirely clear, but my understanding is that it means that "placed" includes all ways one or more counters may end up on a permanent (including the initial placement from entering the battlefield).
Thus, doubling the counters will trigger the ability as long as there were any counters in the first place (i.e. doubling zero counters does not place counters).
To double any type of counters, you place an additional number of counters on the permanent equal to the number of counters that are already on the permanent.
By definition, to "add" a counter, is to place an additional counter. Thus, to place an additional counter, is to "add" a counter.