Herein, we prepare a polythioether with double "clickable" alkyne groups in each repeating unit via an extremely rapid polymerization method and combine it with Cu(I)-catalyzed azide-alkyne cycloaddition (CuAAC) "click" reaction to construct new topological polymers. A monomer having both electron-deficient and electron-rich dual alkyne functionalities was synthesized and subsequently reacted with 1,6-hexanedithiol (HDT) using 1,5,7-triazabicyclo[4.4.0]dec-5-ene (TBD) as a catalyst, and the reaction proceeded at room temperature for 1 min to give polythioether P1. The obtained polymer was then used as a polymer platform and reacted with a variety of azide compounds to modify the pendant alkynes via CuAAC reaction. The post-polymerization modification (PPM) studies have also been shown to be rapid as the reactions reached completion in 1 h. All synthesized polymers were analyzed using various spectroscopic methods and the results confirmed that quantitative "click" reaction efficiency was achieved in each reaction. It is believed the presented strategy not only provides a time-saving approach for both polymer synthesis and PPM, but also brings a new insight to impart the desired functionalities to a polymer chain with CuAAC "click" chemistry. Thus, a new type of polymer structure that can be prepared rapidly and undergoes fast PPM has been introduced to polymer chemistry with this study.