Fast-degrading linear and branched polyanhydrides are obtained by melt-condensation of novel di- and tri-carboxylic acid monomers based on oleic and undecylenic acid synthesized using photoinitiated thiol-ene click chemistry. (1)H NMR spectroscopy, size exclusion chromatography, differential scanning calorimetry, thermogravimetric analysis, and FT-IR spectroscopy have been used to fully characterize these polymers. The hydrolytic degradation of these polymers was studied by means of weight loss, anhydride bond loss, and changes in molecular weight, showing fast degrading properties. Drug release studies from the synthesized polyanhydrides have also been conducted, using rhodamine B as a hydrophobic model drug, to evaluate the potential of these polymers in biomedical applications.