Hops need to stay cold and away from oxygen. In a vacuum tight bag and in the freezer you can go many months to even a few years. LME is best stored cool as well, and in a fridge you can probably expect a few months out of it, but some people even freeze it if need be for longer periods. DME as long as it's stored in an air tight container in a cool dry place can last a long time, probably well over a year. Priming sugar can be stored pretty much indefinitely in a dry or air tight place.
Once you get into grains, well, if they have already been crushed you'd want to use them as soon as you can or otherwise store them in an airtight and cool to cold location to extend their freshness. Uncrushed grains can be stored for much longer, again, with the whole cool and dry theme.
Yeast of both types usually have an expiration or a best use date on them. Liquid shouldn't be frozen, so you should use it before the date listed on the vial or packet. Usually around 3 months after production. Dry yeast can last much longer, but they too usually have a date imprinted on the package which usually means it's good for a couple of years.