What is the allowable span of 2 stacked 6x6's?
I'm sure that will be fine.

The American Wood Council has a free load table table on their website.

If you assume a conservative 1400psi allowable bending stress, a single 6x6 with 12' span can support about 2000 lbs. So your double 6x6 can support at least 4000 lbs (possibly more if you do a good job bonding the two members together so that there's no slip between the faces).

If you have 3 swings, you should assume that each one is packed. Let's say each individual swing has 3 kids at 150 lbs each, for a total of 1350 lbs. [You want to design for the absolute worst case. I bet 3 kids can pile onto one swing.] So you're not that close to the swing limit. You could hang your car from those posts and still have a safe structure.

If you take a look at a swing set or jungle gym at a local park, I'm sure you'll find much skimpier designs.

While I'm quite confidant that your double 6x6s will properly support the bending caused by the kids, you should make sure the rest of the structure is adequate. You didn't ask about this in your question so maybe that means you have it all figured out, but you should be careful not to overlook:


The vertical supports. As Michael Karas says, there will be considerable lateral momentum from 3 kids swinging.
The connection between the horizontal piece and its supports. Especially since you seem to be designing this for use by active kids and not just toddlers, you want to make sure that the beam doesn't fall off the supports with a lot of synchronized swinging. Consider metal brackets (e.g. "hurricane ties") to strengthen that joint.
With 3 swings there will be a lot of momentum and I think you're going to want the whole swing set anchored to the ground. You definitely don't want the whole thing tipping over. Consider that a single 6x6x12' post weighs about 100 lbs. You don't want that falling on a kid.


EDIT: As The Evil Greebo points out in the comments below, the swinging motion will create additional force on the beam—as much as 3x the weight, if the kids can get up to 90˚. I'm still more than comfortable with the structure; while it's conceivable that three chubby kids could pile on a single swing, I seriously doubt they could all swing that high in unison.