Here is an example of an end that exists even though the indexing category is big. Let $\mathcal{S} = \textbf{Set}$, and let $H = \mathcal{S}(-, -) : \mathcal{S}^\textrm{op} \times \mathcal{S} \to \mathcal{S}$ be the hom functor. I claim $1 \cong \int_{X : \mathcal{S}} H(X, X)$ Indeed, by the end form of the Yoneda lemma (or direct verification), it is known that $\textrm{Nat}(F, G) \cong \int_{c : \mathcal{C}} \mathcal{D}(F c, G c)$ for all functors $F, G : \mathcal{C} \to \mathcal{D}$; but the ordinary Yoneda lemma says $\textrm{Nat}(\textrm{id}_{\mathcal{S}}, \textrm{id}_{\mathcal{S}}) = \{ \textrm{id}_{\textrm{id}} \}$ so the claim follows. I think $\mathcal{S}$ can be replaced with any well-pointed topos in this argument.
Of course, there are also ends that don't exist. Let $\mathcal{C}$ be the disjoint union of $\kappa$ copies of a non-trivial group $G$, where $\kappa$ is the size of the universe, and let $K = \mathcal{C}(-, -) : \mathcal{C}^\textrm{op} \times \mathcal{C} \to \textbf{Set}$ be the hom functor again. This makes sense because $\mathcal{C}$ is locally small by construction. Since $\mathcal{C}$ is very nearly discrete, it is easy to compute $\textrm{Nat}(\textrm{id}_\mathcal{C}, \textrm{id}_\mathcal{C})$: it is just the $\kappa$-fold power of $G$. In particular, it is not a small set, so the end of $K$ does not exist.