The cyclooctyne‐functionalized alcohol (1R,8S,9S)‐bicyclo‐[6.1.0]non‐4‐yn‐9‐ylmethanol (BCN‐OH) is applied as initiator for the organo‐catalyzed ring‐opening polymerization (ROP) of morpholine‐2,5‐diones based on the l‐amino acids valine, isoleucine, and phenylalanine. The ROP is catalyzed by a binary system of 1,8‐diazabicyclo[5.4.0]undec‐7‐ene (DBU) and 1‐(3,5‐bis(trifluoromethyl)phenyl)‐3‐cyclohexylthiourea (TU) applying a feed ratio of [M]/[I]/[DBU]/[TU] of 100/1/1/10. Kinetic studies reveal that BCN‐OH is capable to initiate the polymerization of morpholine‐2,5‐diones, which proceed in a controlled manner until monomer conversions of 80%. Characterization by means of 1H NMR spectroscopy, size exclusion chromatography (SEC), and matrix‐assisted laser desorption/ionization‐time of flight‐mass spectrometry confirm the covalent attachment of the cyclooctyne moiety as α‐end group of the poly(morpholine‐2,5‐dione)s with maximum dispersities of 1.25. As a proof of concept, a vitamin A end‐functionalized poly(2‐ethyl‐2‐oxazoline) is coupled to a poly(ester amide) by strain‐promoted azide‐alkyne cycloaddition. Characterization of the block copolymer by SEC and DOSY NMR spectroscopy confirm the successful attachment of the two building blocks. The versatile cyclooctyne moiety shall facilitate a metal‐free attachment of other polymer blocks, targeting ligands or dyes at the α‐end group of well‐defined poly(morpholine‐2,5‐dione)s. In consequence, the approach provides access to a new generation of functionalized poly(ester amide)s, which can be customized for specific needs.