We have geothermal and I'd like to add insulation to my cathedral ceiling, wall, etc. I've heard about "blown in" insulation, but I'm not sure what's the best, safest type to use. I'm worried about out-gassing of chemicals, moisture build-up, etc. Any help would be greatly appreciated.

Here is an idea. Nudura has what is called the Integrated Series which is 4' x 8' sheets of EPS foam that range from 2 5/8" thick to 3 7/8" thick with a shiplap joint. The sheets have wood strapping @ 16" OC built in to them. (for attaching drywall, wood etc) You can also get the sheets with the plastic strip. You can go to nudura.com and check it out. I hope this helps?

Blown-in is an entire class of insulations.

Cellulose (essentially recycled paper/denim (cotton) in many cases)
Fiberglass (like bat insulation but loose).
Rock wool.

Most of them have some sort of borate treatment (generally boric acid) to repel insects and resist fire.

What they do if you're not stripping down to stud bays and repacking from scratch is pop holes in the surface of the wall/ceiling and blow the stuff in and multiple points along each stud bay.
For walls, patching is relatively easy.

On a ceiling...I dunno. You'll want to talk with your insulation contractor about that.

If you're dry-blowing, the installer needs to take care to make sure it fills all voids.
Damp-packing it generally helps it pack down and fill voids somewhat more easily.