New York State Building Code does not allow having more than 2 layers of shingles on any roof.
It would be a code violation to add a 4th layer to this structure and you will not be able to secure permits to do this job.
See section R907.3 of the NY State Residential Code
"New roof coverings shall NOT be installed without first removing existing roof coverings where any of the following conditions occur:
1. Where the existing roof or roof covering is water-soaked or has deteriorated to the point that the existing roof or roof covering is not adequate as a base for additional roofing.
2. Where the existing roof covering is wood shake, slate, clay, cement or asbestos cement tile.
3. Where the existing roof has two or more applications of any type of roof covering."
You can see for yourself the text of the NY State Residential Code here in Chapter 9:
Appear you have ALL
of the above negative conditions present and no way would you be permitted or pass inspection for reroofing if you add another layer.
Should you add another layer, you will find yourself subject to fine, license revocation and damages...and be liable to correct any deficiencies in the roof system at your own expense.
Either convince this customer they need to have the roof done properly or walk away from this job.