A big problem with recycling polymers, (overlooking all the other problems such as obtaining a feedstock of high quality in large enough quantities at a good price, and also having a market for the output) is that they degrade. This is not unique to the recycling process: any thermomechanical processing of polymers will result in some degradation. The resulting polymer is not as strong as the virgin material from which it was made.
One option to offset this is to rebuild the polymer. DSM has a chain extender available (with more in the pipeline). The chain extenders are bifunctional materials that react with the endgroups of the polymer. If both groups can react with a polymer chain, then a new longer chain is formed. Getting two engroups to react is always a challenge of a statistical nature. The chain extender is a small molecule so it can diffuse relatively quickly and find an endgroup to react with. But that is the easy step. Finding a second endgroup is the issue. Since polymers can't move quickly, it takes quite some time for another endgroup to diffuse and find the chain extender that is already bonded to the end of a chain.
An additional challenge can be posed by the actual chemistry of the polymer.
As expected, the chain extender has to be matched appropriately to the base polymer and more specifically to the endgroups of the chain. Even that may not be as easy as you would expect. PET is most often made by copolymerizing ethylene glycol and tererphthalic acid, resulting in -OH groups on both ends of the polymer (albeit in one case as part of a carboxylic acid group). However, DuPont also has a process that copolyermizes dimethyl terephthalate and ethylene glycol, in which case there is a methyl endgroup on one end of the polymer chain. That methyl group certainly won't react with a chain extender, or much of anything else for that matter. This then means that only half the chain endgroups can react with the chain extender - a real challenge.
PET and nylons are good candidates for this because of the potentially reactive engroups. Other condensation polymers, such as urethane and ureas would be candidates, but given the diversity of urethane chemistry that exists along the chain between the urethane groups, recycling urethanes will likely never happen.