What size chain is typically used when hanging a porch swing?
A standard double loop chain is often used for porch swings.
Assuming three hefty occupants (~250 lbs each) and a 100 lb swing, you're up to a load of 850 lbs. If you use 4 lengths of chain to connect the swing, that's 212.5 lbs per chain. So you need at least 2/0 chain, which is rated to a working load of 255 lbs and an UTS of 1020 lbs.
EDIT: If you're a nerd/engineer like me, you can get more technical and take into account angle of the chains from vertical and normal acceleration from swinging. Then your equation comes to:
which will be a bit more than 255 lbs if you're swinging very fast, or have very short chains or a very wide swing. In that case, you may want to go with 3/0 chain.
Pick out the strongest chain that's in your budget and matches your desired aesthetic, then make sure it's no weaker than Doresoom's answer recommends. There's no reason to skimp on vertical supports. Maybe four 300 pound guys are all going to sit down at the same time one day.