I think there is such a thing, but I think it's nurtured rather than being anything inherent. The addictive personality is that way because he or she has a need - love, security, attention or whatever - that hasn't been met at some early point in their life, hence they seek it through the bottle, the weed, the credit card etc.
I think there is some scope for having an addictive personality... wasn't some research done on this a few years back? Wikopedia might have something...
Anyway, yes there's scope for having a personality that is more disposed to becoming addicted to things but I don't think it's as simple as nature. As Jag's says I think there would have to be a trigger for it.
Is it heredity... well yes, possibly it could be I expect. Depression seems to run in families so I guess this could too.
i have read that people do have addictive genes,i believe it is hereditary,some 1 in my family had a problem and it was inhererited by other members of family,enviroment must account for some of it though,it stand to reason