Yes, but you won't like what will probably happen (this is a worst case scenario, hopefully you will avoid most):
The purpose for a building permit is to ensure meeting the minimum standards for construction practices and minimum safety standards. A contractor who works without a permit, does so because they know they are not in compliance (and to save money at your expense). They knew they should have had inspections and permits for their work. Who ever built the addition should be reported to both the building official as well as the Better Business Bureau. It is ultimately the homeowner who is responsible for ensuring the proper permits have been applied for, though.
So when you apply for your permit, you will be pretending as if the work is not completed (you do not hide this fact, you just have to follow the correct process as if it hadn't been built). Your first stop will be with zoning; can you even add an addition, do you have the proper clearances from the side, front and rear property lines. If it is a bedroom, does the septic system (perk test) support an additional bedroom. If your building already meets or exceeds the amount of building allowed on the site or if you do not have the clearances required from the property line, your addition may be required to be removed. There are appeal processes and variance requests you can try before tearing down the addition (Get an architect).
If your zoning review is fine, next you go to permitting. Here you will submit plans (drawings) of what was built. If you do not have these plans, consider hiring an Architect to generate As-Built drawings for this use. Hopefully the plan review comes back with no changes, or you will alread know your addition is not in compliance and may face rebuilding. Depending on the type of construction, your zoning and your local building requirements, you will be required to have inspections of your foundation / footings, the framing, the electrical, mechancial and HVAC systems, etc. affected by the work.
This may require digging the ground back up so the inspector can confirm foundation depth, size and draingage requirements. The interior wall finishes (gypsum board, panelling, etc) may have to be removed in some or all areas so the framing and electrical can be inspected (If one area fails, be prepared to pull all areas down). At each inspection, if the work is found to be lacking, then you will have to correct the work before getting permission to use the room. If there is an electrical or safety violation found, it is possible the Building Official could declare the entire property inhabitable until the offenses are corrected (IE you are homeless until it is fixed).
As you can see, you have to hope beyond belief that the builder constructed everything correctly and that the building officials will work with you to minimize the amount of deconstruction necessary to inspect the work.
Also, you will be charged all the fees associated with plan review and permitting, and you will be charged for each inspection visit (as your builder would have been charged initially had they followed the law).
As for value, here is the real concern: If your home burns down or faces some similar disaster, your home owners insurance will balk at paying; they will blame the illegal construction as the cause. As for the value of your building, not having a permit will make any buyer have a difficult time getting their own insurance, thus harder to sell. The room itself will add value to your property, if it isn't a hazard (IE permitted).
Also, taxes are based upon assessments, which use the land records. Building without a permit, can be seen by local officials as an attempt to avoid paying property taxes, since the land records do not show the addition. Until the official tax records reflect an accurate statement of your building, you may face fines, tax fees and other costs associated with the improvement depending upon how long it has been there unreported.
You may wish to contact a local, licensed Architect who works with the local building department. They will know the personel, know which forms you need to fill out first and how to protect you from an overzealous Building Official (there are exceptions and options within the Code that the Building Official may forget or ignore that an Architect can request be used to prevent tear down or damage). Next time you go to build get the Architect first to protect yourself from what this construction firm did to you.