let X =Hawaiian earring, consider CX be the cone with base X. Now take two coies 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 V 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 the fundamental group of K is not trivial.(a closed path oscillating back to forth from Y to Z around the decreasing circles is not null homotopic. You can get the reference form the paper- "THE COMBINATORIAL STRUCTURE OF THE HAWAIIAN EARRING GROUP" - bY J.W.CANNON & G.R.CONNER )
So now if you consider the inclusion map from CZ or CY to K then for give any covering space of K you can lift CZ or CY (by map lifting theorem). so it follows that any connected covering space of K is homemorphic to K. so K is its own universal cover. and fundamental of K is not trivial.