The Van Kampen theorem implies that, given two path-connected (pointed) topological spaces $(X,p)$ and $(Y,q)$, we can relate the fundamental group of their wegde sum with both their fundamental groups: $$\pi_1(X\vee Y,p\vee q)\cong\pi_1(X,p)\ast\pi_1(Y,q).\qquad(\star)$$ Here, $\ast$ means the free product of groups. Note that the previous holds if we require the wedge point $p\vee q$ to have a simply-connected neighborhood in $X\vee Y$. This always happens if both $X$ and $Y$ are locally simply-connected.
Now, I am trying to construct a counter-example of the fact $(\star)$. For this, I am looking at a space that is not locally simply-connected: the Hawaiian earrings. I denote it as $H\subset{\bf R}^2$, and I take the basepoint $p\in H$ to be the “wild” point $(0,0)$. Now, I take $Y$ to be a circle of radius $2$ and centered at $(2,0)$ in ${\bf R}^2$, and I also take its basepoint $q=(0,0)$. Surely, the wedge sum $(H\vee Y,p\vee q)$ is not locally simply-connected, for the wedge point does not have a simply-connected neighborhood.
My question is: Does that provide a counter-example? My reasoning is as follows:
$\;(1)\;$ The wedge sum is homeomorphic to $H$.
$\;(2)\;$ In that case we have that $\pi_1(H\vee Y,p\vee q)\cong\pi_1(H,p)$.
$\;(3)\;$ We see that $\pi_1(H,p)\ast{\bf Z}\not\cong\pi_1(H,p)$.
I am unsure about steps $(1)$ and $(3)$ above. For $(1)$, a homeomorphism would be given by shifting every circle once to the left. For $(3)$, this is because we're adding a free generator to the (infinite) presentation of $\pi_1(H,p)$? But That does not convince me, since the free group on countably generators has the property that is is invariant under adding countably many free generators.
Please correct me wherever I am wrong!
Unfortunately I cannot answer the question whether $\pi_1(H,p)\ast{\bf Z}\not\cong\pi_1(H,p)$. The fundamental group of the Hawaiian earring is a fairly complicated object as you can see in
I guess that $\pi_1(H,p)\ast{\bf Z}\cong\pi_1(H,p)$, but I do not know how to prove it.
Anyway, a counterexample to $(*)$ can be found in Fundamental group of wedge sum of two coni over Hawaiian earrings. This example is closely related to the Hawaiian earring. It shows that the wedge of two contractible spaces can have a highly non-trivial fundamental group.
Update.
Although the Hawaiian earring behaves "locally bad" at $p = (0,0)$, it is nevertheless true that $$\pi_1(H \vee Y, p \vee q) \cong \pi:1(H.p) * \pi_1(Y,q) . \tag{1}$$ The reason is that $Y$ behaves "locally nice " at $q$. If one of spaces involved in the wedge is "locally nice" at its basepoint, we can prove $(1)$.
See the answers to Under what conditions $\pi_1(X \vee Y) \approx \pi_1(X) * \pi_1(Y)$?