What are the tightest grout joints I can get with natural stone tiles?

That will usually depend on the tile. Marble, granite and limestone tiles are usually cut from blocks and then turned into tiles on a tile saw. If produced by a good tile factory, these tiles will have a size variance that will allow for a 1/16″ grout joint. Imported slate, and other stones, such as quartzite and sandstone, are usually produced with a size variation that will only allow joints from 1/8″ to 3/8″, depending on the stone and factory.