What species of goldfish shouldn't be kept together?

Any kind of goldfish will mix OK with any other kind of goldfish. Goldfish are all one species with many 'sports' that have been developed by selective breeding. The only No! No! is if you have some well bred fish and don't want them to cross breed at breeding time. Then you will need to keep type with type, otherwise, since they are the same species, they will cross breed and your resultant fry will be almost worthless crossbred goldfish.