What are typical resins used in carbon fiber for aircraft? Are any resins impossible to recycle?
This will give you some detail on thermoset resins, which solidify through chemical reaction. In summary:


Epoxy is the primary resin used in aerospace, at least for structures, as it is strong and wear resistant. 
Phenolic is the primary non structural resin used in aerospace, as it is fire resistant. This is important in aircraft interiors.
Polyester is cheaper than epoxy but weaker, so it shows up in less structural or weight critical applications. Boats might use this.
Vinyl ester is somewhere in between epoxy and polyester, so not as good as either. I have never used this. 
Bismaleimide (BMI) is used for high heat applications where maintaining strength is important. This is a slightly less expensive member of the Polyimide family, used for similar purposes.
Cynate ester apparently has military applications, with which I am not familiar.
PEEK as suggested in comment, is the only thermoplastic in this list, which is formed by melting and solidification as by injection molding. It can be recovered for reuse with reduced mechanical properties.


Thermoset resins are generally not recyclable. When composites are intended for recycling the fibers are the object, not the resin. The resin has to be burnt or dissolved away, creating an emissions problem that needs to be dealt with.