We propose a versatile approach to the production of hyperbranched polymers with high degrees of branching and low dispersity values (D), involving slow monomer addition of a thiol/yne monomer to multifunctional core molecules in the presence of photoinitiator and under UV irradiation. The small thiol/yne monomer was synthesized via 1-ethyl-3-(3-(dimethylamino)propyl)carbodiimide hydrochloride (EDC center dot HCl) esterification, and batch polymerizations were performed at varying concentrations. The batch thiol-yne polymerizations had fast reaction kinetics and large dispersity values that increased with increasing concentration. Introduction of monomer by slow addition to a multifunctional alkyne core (tri(prop-2-yn-1-yl) 1,3,5-benzenetricarboxylate) or alkene core (triallyl 1,3,5-benzenetricarboxylate) was found to lower dispersity at monomer concentrations of 0.5-2.0 M. Degrees of branching were determined by H-1 NMR spectroscopy to be greater than 0.8 in most cases. Increasing the fraction of core molecule was found to decrease dispersity to values as low as 1.26 and 1.38 for the alkene core and alkyne core, respectively, for monomer concentrations of 0.5 M with 10 mol % core molecule. Molecular weights of the hyperbranched structures were also determined by light scattering size exclusion chromatography (SEC) detection, and intrinsic viscosities were determined by viscometry SEC detection. The Kuhn-Mark-Houwink-Sakurada alpha parameter decreased from 0.35 for the batch process to values as low as 0.21 (10 mol % alkene core) or 0.16 (10 mol % alkyne core), indicating that the thiol-yne structures became more globular and dense with the slow monomer addition strategy. This simple and versatile approach is a promising new development for the design of hyperbranched polymers of well-controlled molecular weight and molecular weight distributions, with very high degrees of branching.