What proportion of proteins require chaperone-assisted folding?
I'm reminded of a lovely review in Trends in Biochemical Sciences that discusses chaperone independent, partially dependent, and fully dependent proteins in prokaryotes.  The conclusion was that smaller polypeptides are less likely to require chaperone assistance. This is their figure:



That principle still holds to a certain extent (in prokaryotes), but folding assistance is now more broadly understood to include much more than just specific proteins identified as molecular chaperones. The Anfinsen postulate (that the final tertiary structure of a protein depends only on its primary structure) may still hold for small globular proteins, but folding in vivo is almost certainly always assisted. Since you're coming from the perspective of a chemist, don't think of it as a reaction between two molecules in solution. It's a reaction in a gel packed with complex sugars, lipids, proteins, and nucleic acids. 

If you're skeptical of the perspective of a crowded cytoplasm, read the review I linked. It's an important but often under-appreciated aspect of in vivo biochemistry. There is much more to read about it. You might try looking up Allen Minton (AP Minton) and macromolecular crowding in your favorite database.