Extra Cost: <$500
For normal building applications non-toxic drywall is prevalent everywhere at nearly any building materials store and is no more expensive. For building applications near moist conditions the right drywall is more expensive, but well worth the investment.
What: For normal building, simply ask for untreated, "normal" drywall, with no mold-resistant chemical treatments or fungicides. When building for moist conditions (ex. drywall covering a concrete wall or in a bathroom), use fiberglass drywall.
Why: Mold-resistant chemicals and fungicides are highly toxic and can evaporate into your living environment over time. Since drywall is probably the largest surface area in your house, you want to make sure you don't have toxins in it. For moist conditions, fiberglass drywall is naturally mold and fungus-resistant, and also toxin-free.
How: USG, the largest drywall manufacturer in the world, distributes through Home Depot and almost any other building supply store. They have a chemical-free product, you just have to ask for drywall without fungicides or other treatments. For moist applications, the best fiberglass product on the market based on my research, is a product called DenseArmor Plus. This product is great for use near concrete because it is also breathable and hence doesn't trap moisture behind your walls. For bathroom applications, I recommend DenseShield, which is also fiberglass-based, but does not breath, and hence keeps moisture in your bathroom, instead of letting it into your wall cavities.