Aliphatic polyesters are the most common type of biodegradable synthetic polymer used in many pharmaceutical applications nowadays. This report describes the ring-opening polymerization (ROP) of l-lactide (L-LA), ε-caprolactone (CL) and glycolide (Gly) in the presence of a simple, inexpensive and convenient PEG200-BiOct3 catalytic system. The chemical structures of the obtained copolymers were characterized by 1H- or 13C-NMR. GPC was used to estimate the average molecular weight of the resulting polyesters, whereas TGA and DSC were employed to determine the thermal properties of polymeric products. The effects of temperature, reaction time, and catalyst content on the polymerization process were investigated. Importantly, the obtained polyesters were not cyto- or genotoxic, which is significant in terms of the potential for medical applications (e.g., for drug delivery systems). As a result of transesterification, the copolymers obtained had a random distribution of comonomer units along the polymer chain. The thermal analysis indicated an amorphous nature of poly(l-lactide-co-ε-caprolactone) (PLACL) and a low degree of crystallinity of poly(ε-caprolactone-co-glycolide) (PCLGA, Xc = 15.1%), in accordance with the microstructures with random distributions and short sequences of comonomer units (l = 1.02–2.82). Significant differences in reactivity were observed among comonomers, confirming preferential ring opening of L-LA during the copolymerization process.