An example of a space that is not semilocally simply connected is the following: Let $$HR=\bigcup_{n\in\BN}\left\{(x,y)\in \BR^2\,\bigg\vert\,\left(x-\frac{1}{2^n}\right)^2+y^2= \left(\frac{1}{2^n}\right)^2\right\}$$ endowed with the subspace topology. Then $(0,0)$ has no simply connected neighborhood. Indeed every neighborhood of $(0,0)$ contains (ever diminshing) homotopically non-trivial loops. Furthermore these loops are homotopically non-trivial even when considered as loops in $HR$ .

Figure: The Hawaiian rings

It is essential in this example that $HR$ is endowed with the topology induced by its inclusion in the plane. In contrast, the same set endowed with the CW topology is just a bouquet of countably many circles and (as any CW complex) it is semilocaly simply connected.