Let $$M$$ be a Riemannian manifold. Let $$\mathrm{inj}_M(p)$$ be the injectiviy radius at a point $$p\in M$$, which is defined as the biggest $$R>0$$ such that $$\mathrm{exp} \colon B_R(p) \rightarrow M$$ is a diffeomorphism. Then, the injectivity radius of $$M$$ is defined as $$\mathrm{inj}_M :=\mathrm{inf}\{\mathrm{inj}_M(p) : p\in M\}.$$ It is clear that if $$M$$ is not complete, $$\mathrm{inj}_M=0$$ since you can pick a sequence $$\{q_n\}_{n\in \mathbb{N}}$$ which converges to the point $$q$$ where the completeness fails and $$\lim_{n\to \infty}\mathrm{inj}_M(q_n)=0$$. Thus, $$\mathrm{inj}_M=0$$.
Is there any easy example of a complete Riemannian manifold $$M$$ with $$\mathrm{inj}_M=0$$?
This $$M$$ should be non compact. Because if $$M$$ is compact, then $$\mathrm{inj}_M>0$$ since the infimum turns minimum and $$\mathrm{inj}_M(p)>0$$ for each point $$p\in M$$ because the exponential map is a local diffeomorphism.
Consider a surface of revolution such as $$M = \{x \in \mathbb{R}^3 : x_1^2 + x_2^2 = \frac{1}{1+x_3^2} \}$$ with Riemannian metric inherited from the ambient space. For points with large $$|x_3|$$, the injectivity radius becomes arbitrarily small.