Here are some pretty examples of self covering manifolds: Suppose that $F$ is a manifold and $f \colon F \to F$ is a periodic homeomorphism of period $k$. We define $M_f = F \times I \,/\, f$ to be the mapping torus with monodromy $f$. That is, form the product $F \times I$ and identify the two ends via $f$. Note that $M_f$ is an $F$-bundle over the circle.

Now, the $k+1$-fold cover of $M_f$, obtained by "unwrapping the circle direction" is the mapping torus for $f^{k+1} = f$. Thus the $k+1$-fold cover is homeomorphic to $M_f$.

As a concrete example, the trefoil knot complement $X_T$ is a once-punctured torus bundle over the circle, with monodromy of order six. Thus $X_T$ seven-fold covers itself. (And also five-fold covers itself.) This trick works for any torus knot complement.

Here are some pretty examples of self covering manifolds: Suppose that $F$ is a manifold and $f \colon F \to F$ is a periodic homeomorphism of period $k$. We define $M_f = F \times I \,/\, f$ to be the mapping torus with monodromy $f$. That is, form the product $F \times I$ and identify the two ends via $f$. Note that $M_f$ is an $F$-bundle over the circle.

Now, the $(k+1)$-fold cover of $M_f$, obtained by "unwrapping the circle direction" is the mapping torus for $f^{k+1} = f$. Thus the $(k+1)$-fold cover is homeomorphic to $M_f$.

As a concrete example, the trefoil knot complement $X_T$ is a once-punctured torus bundle over the circle, with monodromy of order six. Thus $X_T$ seven-fold covers itself. (And also five-fold covers itself.) This trick works for any torus knot complement.