The art of blowing up protein

Folding? The ability of a protein to fold unaided does not make it an enzyme (a self-epimerase), many proteins do this trick and nobody claims they are enzyme for that reason; where this the case, wild type GFP does not fold unaided anyway (superfolder GFP does).
Autocatalysis? The maturation of the chromophore is in two step, the first is a dehydration. This is a result of a nucleophilic attack of the G67 nitrogen on the S65 carbonyl carbon, which is a proper chemical reaction that is accelerated due to the stressed chi angle of glycine. However, the substrate/product is itself, which probably violates most definitions of enzymes. If that counted, autocatalytic cleavage would make some proteins enzymes. Isoaspartate formation or aspartate-proline cleavage is a tolerated isomerism that happens when a glycine/proline residue follows an aspartate, but is not a promiscuous version of the latter as it's not catalysed. In the second step, the oxidation of GFP's imidazolinone ring system converts an oxygen to a oxygen radical: a substrate and product.
Single turnover? A single turnover enzyme does not obey Michaelis–Menten kinetics, but a second order reaction kinetics. So one could sensibly argue that it is not an enzyme. However, the yeast enzyme thiazole synthase (Thi4p) involved in thiamine synthesis, can do a single turnover, yet it is considered an enzyme. This enzyme is not the only one as several other cofactor enzymes appear to be single turnover.
Cofactor? One way to justify that GFP is not a suicide enzyme is by the fact that oxygen is technically a cofactor.
Overall, what this shows is that GFP is an amusing corner case and that the definition of enzyme is not as black and white as it seems.