Example of variety with big automorphism group? (Big as in, not a variety in a reasonable way.) Let $X$ be some algebraic variety over $k$. In some situations, like $X = P^1_k$ (where it some quasi-projective variety), the set of automorphisms has a natural algebraic structure. 
(If $X$ is a smooth genus one curve, then I think $Aut(X)$ also has a natural algebraic structure: it contains a copy of the elliptic curve $E$, corresponding to translating and changing the base point, $t_g$, and then some finite set of automorphisms $F = Aut(E,e)$. The finite set of automorphisms just depends on the j invariant, so the algebraic structure here should look like $E \times F$, where the product is given $t_g f t_{g^{-1}}$, for $f$ in $Aut(E)$ for some fixed basepoint $e$. I'm not sure though.)
In other situations $Aut(X)$ is presumably very large.
Could someone expand my intuition for this subject by giving me an example of a variety where the automorphisms don't reasonably have a variety (or scheme) structure?
(Motivation: Just thinking about why the definition of a group action of an algebraic group is in terms of $G \times X \to X$, not $G \to Aut(X)$. This question is due to toxic exposure to the nlab.)
 A: See this MO question for discussion. It looks like for $X = \mathbb{A}^2$ there are already problems.
As far as your motivation, suppose $C$ is some monoidal category, $m$ is a monoid in it, and $c$ is an object. Then you can always define an action of $m$ on $c$ in terms of an action map $m \otimes c \to c$ satisfying the usual axioms. You might want to define an action as a monoid homomorphism $m \to \text{End}(c)$, but the problem is that $\text{End}(c)$ won't always exist as an object in $C$. It will if $C$ is in addition closed monoidal, but this just isn't always true. For example, $\text{Top}$ is not cartesian closed, and yet I can still sensibly write down what it means for a topological group to act continuously on a topological space. Varieties, or for that matter schemes, are also not cartesian closed. 
In the case of schemes, an action in the above sense is the same thing as a natural family of actions $G(R) \times X(R) \to X(R)$ of the group $G(R)$ of $R$-points of $G$ on the set $X(R)$ of $R$-points of $X$ (and this generalizes to any cartesian monoidal category). So this is a very sensible thing to write down, and among other things is easy and natural to write down in examples, e.g. the action of $GL_n$ on $\mathbb{A}^n$. Whereas the Aut scheme of $\mathbb{A}^n$, if it exists, is presumably fairly nasty. 
