Finally, a feedback loop that works in mankind's favor.
Humans burning carbon to heat houses -> seaweeds grow like crazy clogging up the oceans -> humans use seaweeds to insulate houses, until -> no more carbon is needed to heat houses!

Can anybody figure out what the R value per inch is?

Hi there, here in Ibiza this has been used in traditional houses since many years, and as the island is full of artist, is also used by them, an example of one of the artist living in Ibiza:
http://annedeharlez.blogspot.com.es/2012/07/exhibition-of-photos-in-new-york.html

From a marine biologist's perspective, just a technical point, Posidonia is a sea grass not a seaweed. Sea grass beds in many parts of the world are a threatened habitat. I'm not sure this makes any real ecological sense.

Seaweed has been used in houses as insulation here in Nova Scotia for at least 100 years.

Let's hope that neptune grass is not habitat for some organism that an endangered species eats. :-) I wonder if some sand-cleaning machines I have seen on the beach in the USA could be modified to harvest the neptune grass.

The Posidonia Oceanica oxygenates water, protects the beaches from erosion and is habitat for lots and lots of organisms! It grew up very very slowly. Please do not harvest them!!!! And worst, with some industrial cleaning machines.

Make use of natural products, is almost never an ecological attitude.

More jobs & income for Greece, Italy & Spain for sure & Turkey.

As shrimpfish pointed out, Posidonia is a seagrass, not seaweed. There's a big difference!
About seaweeds... Only a few species of seaweeds (more correctly called macroalgae) are in fact weeds and work in the way that BeWalt describes - clogging up the ocean and resulting in adverse ecological and human health consequences. Seaweeds are integral components of marine ecosystems, and underpin coastal (and even abyssal!) marine foodwebs.
Seagrasses on the other hands are true plants, like land plants - i.e. they are angiosperms. They also form part of marine food webs, but are not always as readily consumed as the macroalgae for various reasons (some megaherbivores do graze them directly, however). Species like Posidonia are very important for habitat creation, providing a substratum for colonisation of algal epiphytes, and enabling the consolidation of marine sediments.
Sorry to get fussy about seaweeds vs. seagrasses, but from ecological and evolutionary perspectives they are vastly different.
Those "neptune balls" form because of lignin and cellulose fibres, a characteristic of seagrasses (and other angiosperms) but not of seaweeds.
Kind regards! AJ (Cynical phycologist)