Herein we report the synthesis of biodegradable hyperbranched polyglycerols (BHPGs) having acid-cleavable core structure by anionic ring-opening multibranching polymerization (ROMBP) of glycidol using initiators bearing dimethyl and cyclohexyl ketal groups. Five different multifunctional initiators carrying one to four ketal groups and two to four hydroxyl groups per molecule were synthesized. The hydroxyl carrying initiators containing one ketal group per molecule were synthesized from ethylene glycol. An alkyne-azide click reaction was used for synthesizing initiators containing multiple cyclohexyl ketal linkages and hydroxyl groups. The synthesized BHPGs exhibited monomodal molecular weight distributions and polydispersity in the range of 1.2 to 1.6, indicating the controlled nature of the polymerizations. The polymers were relatively stable at physiological pH but degraded at acidic pH values. The polymer degradation was dependent on the type of ketal structure present in the BHPG; polymers with cyclohexyl ketal groups degraded at much slower rates than those with dimethyl ketal groups at a given pH. Good control of polymer degradation was achieved under mild acidic conditions by changing the structure of ketal linkages. A precise control of the molecular weight of the degraded HPG was achieved by controlling the number of ketal groups within the core, as revealed from the gel permeation chromatography (GPC) analyses. The decrease in the polymer molecular weights upon degradation was correlated well with the number of ketal groups in their core structure. Our data support the suggestion that glycidol was polymerized uniformly from all hydroxyl groups of the initiators. BHPGs and their degradation products were highly biocompatible, as measured by blood coagulation, complement activation, platelet activation, and cell viability assays. The controlled degradation profiles of these polymers together with their excellent biocompatibility make them suitable for drug delivery and bioconjugation applications.