Why is the category of groups not closed, or enriched over itself? Wikipedia has some cryptic things to say about the non-abelian structure of the category of groups.
It says the category of groups is not additive because "there is no natural way to define the sum of group homomorphisms". What about just adding them pointwise? I understand groups are not additive because they're not $\mathsf{Ab}$-enriched, so pointwise addition would be commutative. But then, aren't groups enriched over themselves at least? I remember reading about a non-abelian tensor product somewhere...
Even if not enriched over itself, is it not at least closed?

It's a consequence of the famous Eckmann-Hilton argument. Let $u,v : G \to H$ be two groups homomorphisms. Define $w : G \to H$ be $w(g) = u(g) v(g)$ as you suggest. Then for $w$ to be a group homomorphism, you need:
$$w(gh) = u(gh) v(gh) = u(g) u(h) v(g) v(h) \\
= w(g) w(h) = u(g) v(g) u(h) v(h) \\
\implies u(h) v(g) = v(g) u(h)$$
and so this defines a group homomorphism iff $u(h)$ commutes with $v(g)$ for all $h,g \in G$. Obviously, this is not necessarily the case, so in general pointwise multiplication doesn't yield a group homomorphism. It's not that "addition" of group homomorphisms isn't commutative that prevents $\mathsf{Grp}$ from being abelian; it's that it isn't even defined.
A lot of work has been done to try and see what properties characterize $\mathsf{Grp}$; as far as I know, semi-abelian categories capture a great deal about $\mathsf{Grp}$.
