The Hawaiian earring is the space $$X = \cup_{i=1}^{\infty}\{(x,y) : (x-1/i)^2+y^2=1/i^2\}.$$
It is different from wedge of infinite circles.
But why can't we apply van Kampen's directly by taking each open path connected set to be $$U_i = X\backslash\{(2/j,0)\}_{j\in\mathbb{N}}^{j\neq i}$$
and hence obtain that $$ \pi_1(X,(0,0)) = *_{\infty}\mathbb{Z} $$ ? Am I missing some detail?