I am flying into ANC in the summer and want to travel the Denali Hwy. Even though it is restricted to virtually all car rental companies I know that people still drive it.

Are there any national chains that permit using this road?

I think you are confused. Most of the the Denali road is restricted to all private vehicles (with some exceptions, none of which are related to rental cars).

You can drive the first (I forget how many -- 10 or 12?) miles of the Denali road in your car, whether you bought, rented, or purloined it. To travel the rest of the road, you'd need to take one of the horrid Denali schoolbuses.

No, I am talking about the Denali Hwy, not the road going thru the park. It is different.

It's not paved and I guess there is a lot of damage to cars from gravel. Also, there are limited services.

Here is a link I found to some information:

http://www.glennallen.ak.blm.gov/DenaliHwy/denali.html