Someone with a tendency to save everything, accumulating more and more, is a hoarder. It can be very difficult for a hoarder to throw anything away.
You can use this word for someone who stockpiles things like food or paper goods in case of emergencies or natural disasters. This type of hoarder is similar to animals that hoard food so they'll have enough to last through the winter. A compulsive hoarder is a person whose hoarding instinct has gotten out of control, leaving them psychologically unable to get rid of anything. This type of hoarder often lives in a home dangerously overcrowded with objects they've acquired.