In Hatcher, they give the following example of a simply connected covering space of $S^1 \vee S^1$ 
. I am having trouble convincing myself that it is simply connected. I suspect it is contractible. However, I am getting lost with the infinity of of steps it would take to contract it.
Might anyone have any hints? Thank you in advance,
Raphaël Fua
Don't worry about showing that the cover is contractible: just show that every loop is homotopic to the constant loop.
For that, let $E$ be the supposed universal cover in your post and let $\alpha$ be a loop in $E$. Since the image of $\alpha$ must be compact, its image must live in a finite sub-tree $T$ of $E$. The point is that $\alpha$ can't be wild and run off to all these strange infinite corners: it's contained in a finite sub-tree. Now, Hatcher proves that finite trees are contractible, so use that to give a homotopy of $\alpha$ down to the constant map. That same homotopy in $T$ is a homotopy within $E$ by inclusions, so you're done.