example of a semilocally simply connected space which is not locally simply connected

Let $HR$ be the Hawaiian rings, and define $X$ to be the cone over $HR.$ Then, $X$ is connected, locally connected, and semilocally simply connected, but not locally simply connected.

Too see this, let $p\in HR$ be the point to which the circles converge in $HR,$ and represent $X$ as $HR\times [0,1]/HR\times \left\{0\right\}.$ Then, every small enough neighborhood of $q:=(p,1)\in X$ fails to be simply connected. However, since $X$ is a cone, it is contractible, so all loops (in particular, loops in a neighborhood of $q$) can be contracted to a point within $X$.