Do non-enzyme catalysed reaction pathways exist?
If by enzyme you mean "protein" aka polypeptide, than there are such things as catalitic RNAs. Those are molecules of RNA that facilitate chemical reactions but don't change themselves (definition of catalyst). I think that, based on the discovery of such RNAs, it is now believed that life might have started from or with the help of catalytic RNAs (please forgive the speculative tone, see self-replicating RNAs).

You might find this article interesting: The Road to Non-Enzymatic Molecular Networks, which describes non-enzymatic networks. The problem with searching this topic is that most current publications, seem to me, to concentrate on way to transfer enzymatic reactions into inorganic catalysis systems--because those systems can be more clean and easy to expand--than purifying enzymes for biotech.