let $X$=Hawaiian earring, consider $CX$ be the cone with base X. Now take two copies of this space named as $CZ$ and $CY$ and $z$ be the point where all circles of $Z$ are tangent similarly for $y$. Now consider $K= CZ \wedge CY$ (wedge sum of two spaces identifying the point $z$ with $y$). Now $K$ is connected and observe that both $CZ$ and $CY$ are contractable, but $\pi_1(K)$ is not trivial. For example, a closed path oscillating back to forth from $Y$ to $Z$ around the decreasing circles is not null homotopic. A good reference for this is J.W.Cannon and G.R. Conner's paper- "The combinational structure of the hawaiian earing group."
Consider the inclusion map from $CZ$ or $CY$ to $K$. For any covering space $\tilde{K}$ of $K$, there exists a lift $CZ$ or $CY$ to $\tilde{K}$ by map lifting theorem. It follows that any connected covering space of $K$ is homeomorphic to $K$, and so $K$ is its own universal cover. However, as mentioned above $\pi_1(K)$ is not trivial.
