This work represents our initial effort in identifying azide/alkyne pairs for optimal reactivity in copper-catalyzed azide-alkyne cycloaddition (CuAAC) reactions. In previous works, we have identified chelating azides, in particular 2-picolyl azide, as "privileged" azide substrates with high CuAAC reactivity. In the current work, two types of alkynes are shown to undergo rapid CuAAC reactions under both copper(II)- (via an induction period) and copper(I)-catalyzed conditions. The first type of the alkynes bears relatively acidic ethynyl C-H bonds, while the second type contains an -(triazolylmethyl)propargylic moiety that produces a self-accelerating effect. The rankings of reactivity under both copper(II)- and copper(I)-catalyzed conditions are provided. The observations on how other reaction parameters such as accelerating ligand, reducing agent, or identity of azide alter the relative reactivity of alkynes are described and, to the best of our ability, explained.