For EU citizens, after five years of residency, one can acquire a permanent oppholdstillatelse with the local authorities in Norway. Is there a particular advantage in getting such a permit other than the possibility of staying in the country without a permanent work contract? Do I gain any rights in Norway by having such a permit?
If the rules in the EEA are similar to those within the EU, you do get additional rights but those are not necessarily contingent on getting the permit (at least that's what the EU suggests but I am told it's not entirely settled). Under this interpretation, you are a permanent resident by virtue of having lived and worked in the country for five years, the permit is just a way to document this/have your rights recognized should you need to.
The main consequence is indeed that you now have an indefinite right to stay in the country and are protected against expulsion, even if you were to lose your job or the financial means to support yourself. You should also have access to welfare benefits under the same conditions than locals.
Since it should be free and not need to be renewed, I would definitely apply for it. You never know what the future holds and even if you can't really see yourself in this position now, if you do need it in the future, it might be too late to apply. For example, if a serious illness befell you and you were to lose your job and need expensive treatment, it would give you the possibility to choose to get health coverage and disability benefits in Norway (at least to the extent that Norwegian law provides for Norwegians of course, which might or might not be better than in your own country – mostly better I think).