Use the adverb perpetually if something is never, ever going to stop doing what it's doing, like your perpetually chattering cousin.
Perpetually comes from the adjective perpetual, which comes from the Latin for "never stopping." Although there might be some things that might exist perpetually, like the universe (hopefully), we usually use perpetually for things that just feel like they're never going to end. If the coffee shop you go to is perpetually playing the same album, bring them a new one. A good synonym is incessantly.