I was asked about this in a text today, and rather than write back a long text, I decided to answer here. This could also be in the FAQ, but I will just link to this post.
Eggs come in all sizes and there are many factors that come into play. The genetics of the breed comes first to mind. A goose’s first “pullet” eggs will still be extra, extra large on the chicken scale. They are big birds and put a lot into each egg, so it needs to be large. Some chicken breeds are known for laying extra large eggs. And some for laying small eggs.
Pullet eggs are those smaller than normal eggs that many hens start with. Especially the early laying breeds and hybrids, some skip this and wait a few more weeks before starting to lay, but start with large eggs. This is also genetic and breed correlated, but some individual pullets will start laying earlier or later than the norm for their breed.
Sometimes a hen (or more often a pullet) will lay an extra large egg that contains 2 yolks. Usually referred to as a “double yolker”, these are essentially mistakes in the bird’s reproductive tract. They usually skip a day before laying one, so you aren’t really getting more yolks, just 2 in 1. Breeders never select for a propensity to lay double yolk eggs because they rarely hatch (there are exceptions – seems like every one is documented on YouTube). Because you are not really getting more egg (due to the skipped day), they are not the bonus they might seem to be. It also seems like it would be hard on the hen to lay the larger than normal eggs.
If you want the largest eggs, be sure to select breeds or hybrids known for laying large to jumbo eggs. Commercial hens are bred to lay a consistent size large, as any other size does not make as much money per pound of feed consumed. The larger heritage breeds are your best bet for the largest possible eggs, or hybrids of those large egg breeds.
Hybrid vigor can also play a role. The largest chicken eggs on my farm right now are from some Olive Egger project hens that are retired from breeding, but still laying and some of them are laying enormous eggs. The really large eggs from these hens no longer hatch well, but they sure bulk up a carton of eating eggs.
As far as I know, breeding for exceptionally large eggs is likely to produce poor hatches and possibly limit the lifespan of the hens, neither are choices I am prepared to make as a breeder.