(On a related note, see this question. )
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.
Set $U_i = X\backslash\{(2/i,0)\}$. For each $i$, $U_i$ deformation retracts to a space that is homeomorphic to $X$. Moreover, $\cap_{i=1}^{\infty}U_i$ deformation retracts to $(0,0)$. Using van-Kampen's, we write
$$\pi_1(X,(0,0)) = *_{\infty}\pi_1(X,(0,0)). $$
Do such recursions make sense in group theory? Can we conclude from the above recursion anything about Hawaiian ring's fundamental group? For instance, that it is uncountable?