Let $H$ be the Hawaiian earring and let H' be the reflection of the Hawaiian earring across the $y$-axis (in the Wikipedia picture). There is a canonical homomorphism from the free product \pi_1(H) * \pi_1(H') to \pi_1(H \cup H') (with basepoint their intersection), but it is not an isomorphism.
This was intended to be a recent homework problem of mine, but as stated the problem actually asked whether the two groups are abstractly isomorphic. I don't know the answer to this question, and neither does my professor. My guess is that they are not isomorphic, but I don't have good intuitions about such large groups.
Edit: to be clear, I know how to do the intended problem, and I also know that H \cup H' is homeomorphic to $H$.