I've been trying to make peace with the fact that countably infinite wedge of circles is so different from the hawaiian earring. Here are some thoughts:

• Topology on the hawaiian earring is very different. Eg: small opens around the center of infinite wedge is contractible, while no small open around the origin of the hawaiian earring is contractible.
• We can take infinite products of group elements in the hawaiian earring, since the radii decrease. For example, we can take a loop at each circle of radius $1/n$ by making it traverse the circle of radius $1/n$ in the interval $t \in [(n-1)/n, n/(n+1)]$, and stay at $(0, 0)$ at $t=1$.