Re: Building a large spire out of blocks of eggcrate, what to do about the sides?
Fuzeit is a polymer construction adhesive that is non-toxic and fume free. You could use that to attach the slate.
Gorilla glue would work as well, but with the scant points of attachment, I'd recommend filling the tower with Great Stuff after it has set to get a really solid bond. That might be a good idea with Fuzeit as well. It would eliminate the potential gaps your critters (or their food) could crawl into.
Just an aside: I keep wanting to see a gold "Trump" of the side of your structure somewhere. Just a joke people- no offense intended.
There's Bears in these here Woods. Wouldn't have it any other way.