The contractor doing my roofing laid the underlayment down over a week ago and hasn’t returned yet to complete the job. How long can the underlayment remain exposed without damage?

Underlayment’s primary enemy isn’t time it’s exposure to the elements. Once it gets wet, wrinkling and buckling are inevitable. If they become too severe, the underlayment will have to be removed and a new one put down before any shingles can be attached.