- Compulsive hoarder (the mental health diagnosis)
- Messy (the adjective used as a noun, plural "messies", I think originating from Sandra Felton's Messies Manual, but I could be wrong.)
I'd say that this list approximately follows the spectrum from most to least extreme. The "messy" may even have a relative normal amount of stuff and just be unable to keep it in order. The compulsive hoarder may hoard multiple homes and end up living in their car. Not all hoarders have an issue with acquisition; pretty much all hoarders have an issue with disposal.
The list also happens to approximately follow the spectrum from least to most commonly known--I suspect that only members of messy support groups use "messy" as a noun describing a person.
There's also "magpie", with the implication that the person flits around and collects shiny things. I wouldn't consider this a synonym of any of the others, but it is a related concept.