I think this question is impossible to answer. There are dozens of chemical types for label adhesives. Most are thermoplastic, meaning they don't form new covalent bonds with the surface. But some are thermosets, meaning they react and cross-link.
In general, it is quite difficult to dissolve a crosslinked polymer since the dissolution process requires destruction (scission) of covalent bonds. The ways to do that are obvious: heat, oxidation, reduction, acids, or bases.
One thing that needs to be said here is consideration of both the opportunity cost as well as the environmental cost of your method generally makes all but the easiest methods inadvisable. When you say the adhesive "does not respond" to X, do you soak the label in X? (I typically use a slightly larger container (say part of a PET bottle) and a shallow layer of the solvent and allow it to soak overnight.
It's always a good idea, in my opinion, to start with water with a drop or two of surfactant in it. If hot water doesn't work, I move on to 70% IsoPrOH and give that a few hrs. Next, I use an alkaline (basic) detergent solution and give that another overnight soak. If it's worth my time, I end with an acetone soak (I have acetone which I reuse for this purpose in its own container.) One of these works most of the time. When none of these work, I toss (recycle) the jar. You may or may not know that not all polymers are soluble. So, your assumption that there is a solvent for the label you want to remove isn't necessarily true.
The most aggressive solvent I am aware of is mixtures of methylene chloride, dodecylbenzene sulfonic acid, and N-methyl pyrrolidone. That was used to de-pot Soviet microelectronic back in the Cold War days to reverse-engineer them. Anyway, glass is mostly Si-O-Si bonds and alkali bases should loosen any organic's adhesion to it.
Be sure to remove any surface layer on the label - abrasion is generally quickest. Oh, speaking of dangerous chemicals (methylene chloride is a carcinogen, and ddbsa is corrosive to the skin), I've also used aqueous HF to clean glass. The HF actually removes some of the silica, but if done carefully enough doesn't visibly damage the glass surface (never would try this with glass jars; why bother? but with expensive glass flasks, and other apparatuses, it was worth the hassle. The disposal of the waste solvent was a pain, though.)