[cypher]

Sorry I don't know what bird berries are!

The best treats are just natural foods, things like sunflower seeds, pumpkin seeds (both in the shell if possible) mealworms are very popular with Russian hams so make a great hand feeding treat, all these are high in fat though so not too many, tiny pieces of walnut, pine nut or brazil nut go down well too, I give mine a monkey nut once a week (make a tiny hole in each end of the shell so he can smell the nuts inside) millet sprays are usually enjoyed too.
A small piece of fresh veg within the safe list each day is a good healthy addition to the diet & can also be hand fed as a treat.
Other seeds that are healthy & liked (again in moderation) are hemp & flax (linseed).
A small piece of plain cooked chicken or dried chicken once or twice a week makes a good high protein low fat treat too.

If you got your ham from a pet shop then he will be a Russian hybrid (cross WW & Campbells Russian) they are prone to diabetes so avoid any sweet treats, sugar, honey or fruit & feed a good dwarf specific mix with no fruit or sugar in it, not sure where you are but in the UK Burgess dwarf hamster harvest is most commonly used.

ETA Amity I wouldn't feed pasta or puffed cereals to a diabetes prone species, a tiny bit now & then is ok but better to stick to healthier treats.