Are there categories whose internal hom is somewhat 'exotic'? The internal hom (or exponential object) is basically a reification of the 'external' hom. It can be defined in any cartesian (or even monoidal, more on this later) category as the right adjoint of the (monoidal) product.
My question is: are there some categories whose internal hom behaves quite unexpectedly? Or even some interesting examples where internalization really deforms the external hom in a non-trivial way.
The question arises from the observaton that the 'external' hom can be assumed to be a set (assume the category to be locally small, or wave your hand hard enough), and thus is already somewhat reified. We expect some 'structure' on it, namely elements and perhaps even subobjects. In a sufficiently rich category (say a topos), this structure is internalizable as well. So it might be the case that some exotic behavior emerge, or some collapse happens.
I expect this to happen mainly for non-Cartesian monoidal closed category, because the monoidal product can be quite convoluted. 
On the other hand, Section 3 of the internal hom page of the nLab seems to prove the internal hom shares some strong properties of the 'external' hom, which might hint to the fact they are really 'the same'.
 A: This answer describes another way in which exotic (fattened up) internal homs can arise, similar in spirit to those above, but different.
In a monoidal closed category the canonical map
$$j:C(X,Y) \to C(I,[X,Y])$$
is invertible, which tells you that the ``elements" of the internal hom are just  morphisms $X \to Y$.
Invertibility of the canonical map $j$ corresponds to invertibility of the left unit map $l:I \otimes X \to X$.
In a skew monoidal category the left unit map, amongst others, is not required to be invertible -- it follows that in a closed skew monoidal category the internal hom $[X,Y]$ should ``contain" all morphisms $X \to Y$ but can contain other stuff too -- for instance weak maps.
There is, for example, a closed skew monoidal structure on the category $SMonCat_s$ of symmetric monoidal categories and strict symmetric monoidal functors in which the hom $[X,Y]$ consists of the non-strict symmetric monoidal functors (those preserving the structure up to coherent isomorphism).  
See https://arxiv.org/abs/1510.01467 for more on this idea.
A: Jakob and Peter have mentioned the category of $G$-sets for a group $G$. This category remains cartesian closed if $G$ is replaced by a monoid $M$, or for that matter by a category (so we are looking at a presheaf category, which is not only cartesian closed but a topos), but IMO it's tricky to guess what exponential objects look like even in the monoid case, since the usual way the answer for $G$ a group is presented involves inverses and it's not at all obvious how to rewrite the answer without inverses so it generalizes. I really do recommend trying to guess the answer; I'm about to spoil it!

For me it's easier to go all the way up to the generality of presheaf categories even just to understand the case of monoids, so let's consider a general presheaf category $[C^{op}, \text{Set}]$, two presheaves $F, G$, and their exponential $[F, G]$. We can probe the exponential by mapping representables into it: if $c \in C$ is an object and we denote by $c$ the corresponding representable presheaf, then we have
$$\text{Hom}(c, [F, G]) \cong \text{Hom}(c \times F, G)$$
and moreover this identification is functorial in $c$ so this is actually a complete description of $[F, G]$ as a presheaf (by the Yoneda lemma; making this remark explicit for searchability). Note that the monoidal unit here is the terminal presheaf with constant value the terminal object $1$ in $\text{Set}$, which is representable iff $C$ itself has a terminal object. Maps out of the unit can be thought of as "global points" or "global sections" of presheaves. 
Let's specialize this to monoids (so $C$ has a single object $c$ with $\text{End}(c) \cong M$ for $M$ a monoid). Here $[C^{op}, \text{Set}]$ becomes the category of right $M$-sets and the unique representable presheaf is $M$ regarded as a right $M$-set via right multiplication (Cayley's theorem!), so the internal hom between two right $M$-sets $X$ and $Y$ is 
$$\text{Hom}_M(M \times X, Y)$$
where $M \times X$ has the diagonal action, and one can check that the right $M$-action on this homset comes from left multiplication on $M$. Did you guess this? I didn't! To be honest I really don't have a good sense of how to think about this construction, although note that taking fixed points of the right $M$-action does produce $\text{Hom}_M(X, Y)$ as expected, since it corresponds to quotienting $M \times X$ by the left $M$-action, which produces $X$. 
The further simplification that occurs when $M$ is a group $G$ comes from the fact that $G \times X$ with the diagonal action is canonically isomorphic to $G \times X$ with the action only on $G$ (equivalently, with the action on $X$ trivialized); the isomorphism is given by
$$G \times X \ni (g, x) \mapsto (g, xg^{-1}) \in G \times X_d$$
where I write $X_d$ for $X$ with the trivial $G$-action. Hence
$$\text{Hom}_G(G \times X, Y) \cong \text{Hom}_G(G \times X_d, Y) \cong \text{Hom}_{\text{Set}}(X, Y)$$
as usual, and then when we compute the $G$-action coming from the left action on the copy of $G$ that has now disappeared we get the usual conjugation formula. There's probably more to say about this isomorphism, having to do with torsors and so forth.
A: I've seen many great positive answers to this question, providing examples of exotic internal homs. In this answer, I would like to show obstructions to the exoticism of the monoidal closed structure. I think that these examples will make this question and all the provided answers even more interesting. Nothing shapes like a boundary.
The most classical result in this direction is the monoidal structure on the category $\mathsf{Top}$ of topological spaces.

Prop. 7.1.1 Handbook of categorical Algebra 2, Borceux shows that the internal hom of any simmetric monoidal structure in Top has to match with the cartesian structure at least on the level of the underlying set.

This result highly depends on the fact that whatever monoidal closed structure $(\mathsf{I}, \otimes, [\_,\_])$ you have on a category $\mathcal{A}$, the $\mathsf{I}$-points of the internal hom recover the external hom,
$$\mathcal{A}(\mathsf{I}, [A,B]) \cong \mathcal{A}(A,B).$$
This observation spots an entanglement between the internal and the external logic of the category that unveil some rigidities of the monoidal structure. 
Along those lines some research has been developed in the direction of showing that there exists some obstruction in admitting a monoidal biclosed structure.

Topological categories with many symmetric monoidal closed structures, by Kelly and Rossi, show that there exist topological categories which admit a proper class of symmetric monoidal closed structures. 
Algebraic categories with few monoidal biclosed structures or none by Kelly, Foltz and Lair, goes in the other direction, proving (among other stuff) the two following theorems.
Prop.  If an equational variety admits a monoidal biclosed structure, every idempotent algebra is self-commuting.
Prop. The categories of magmas, of semigroups, of magmas with identity, of monoids, of groups, of rings, and of commutative rings, admit no monoidal biclosed structures whatsoever; the category of abelian groups admits none but the classical one, and similarly for abelian monoids; and that the category of small categories admits exactly two, each symmetric, one being the classical Cartesian closed structure.

The last paper is very much inspired by the Czech school, among the very influential papers let me mention

  
*
  
*A. Pultr, Extending tensor products to structures of closed categories, Comm. Math. Univ. Carolinae
  13 (1972) 599-616.
  
*A. Pultr, Closed categories of models of Gabriel theories (manuscript, Charles Univ. Prague, 1973).
  
*J. Rosický, One obstruction for closedness, Comm. Math. Univ. Carolinae 18 (1977). 311-318.
  

If you have other examples that prove how having a closed monoidal structure imposes some rigidity on the underlying category, please contribute to this question with a comment.
A: Typically, internal homs of $\newcommand{\C}{\textbf{C}}\C$ will look different from external homs just when “elements/points of $X$” (for objects $X \in \C$) are different from “maps $I \to X$” (where $I$ is the monoidal unit); or slightly more precisely, when $\C$ comes with a canonical forgetful functor $\newcommand{\Set}{\textbf{Set}} U : \C \to \Set$, which is different from the representable $\C(I,-)$.
This is because (as noted in Jakob Werner’s answer) maps $I \to [X,Y]$ correspond to (external) maps $X \to Y$, and so if we want “points of the internal hom” to be different from external arrows, that implies that “points of the internal hom” must be different from “maps from $I$ to the internal hom”.
This idea suggests several examples:


*

*$G\text{-}\Set$, for a group $G$ (as in Jakob’s answer).  Here the monoidal unit is the terminal object $1$, and maps $1 \to X$ correspond not to arbitrary points of $X$ but just to fixpoints of the $G$-action.  So the external maps $X \to Y$ (i.e. $G$-equivariant maps) correspond to fixpoints in the $G$-set $[X,Y]$; arbitrary points of $[X,Y]$ correspond to not-necessarily-equivariant functions $X \to Y$.

*The category of graded Abelian groups $\newcommand{\Ab}{\mathrm{Ab}}\newcommand{\Z}{\mathbb{Z}} \Ab^\Z$, with the graded tensor product $(X \otimes Y)_n = \coprod_{i+j=n} X_i \otimes Y_j$.  (Or graded modules over a ring, or $\mathbf{N}$-graded, etc.)  In this case, the monoidal unit is $\Z$ in degree 0 and trivial in other degrees, and maps $I \to X$ correspond to elements of $X_0$.  So external maps $X \to Y$ correspond to elements of $[X,Y]_0$, and elements of $[X,Y]$ in other degrees correspond to degree-shifting maps between $X$ and $Y$.

*Categories of chain complexes, $\newcommand{\Ch}{\mathrm{Ch}}\Ch(\Ab)$.  Mostly as in the previous case, but now maps $I \to X$ correspond just to cycles in $X_0$, so external maps $X \to Y$ correspond to degree-0 cycles in $[X,Y]$, while arbitrary elements of $[X,Y]_n$ correspond to maps $X \to Y$ shifting degree by $n$ and not necessarily respecting the boundary operator.
Even if these are not as exotic as you were hoping for, hopefully the general principle “look for categories where maps out of the monoidal unit don’t correspond to ‘elements/points’ of objects” may help find more exotic examples.
A: $\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\Set}{\mathrm{Set}}\newcommand{\hom}{\mathrm{hom}}$In this example one cannot really say that the internal Hom is “exotic”, but at least it is morally different from the external Hom.
Let $G$ be a group and let $\Set(G)$ be the category of sets with a $G$-action. Morphisms of this category are mappings of sets which commute with the action. This category has finite products, given by the products of the underlying sets. In fact, it is cartesian closed: The internal Hom $\hom(X,Y)$ is the set of all mappings $X \to Y$, with the $G$-action given by conjugation: $$\sigma \cdot f := X \xrightarrow{\cdot \sigma^{-1}} X \xrightarrow{f} Y \xrightarrow{\cdot \sigma} Y. $$
Note however that in a monoidally closed category, the external Hom can always be recoverd from the internal Hom as the set of its global elements:
$$ \Hom(X,Y) \cong \Gamma(\hom(X,Y)) := \Hom( I, \hom(X,Y)).$$
Here, $I$ is the unit object of the monoidal category.
In the above example, global elements of some $G$-set $Z$ are morphisms from the one-point $G$-set to $Z$. These correspond precisely to the fixed points of $Z$ under the $G$-action: $\Gamma(Z) = Z^G$. In particular, the global elements of the internal Hom are mappings invariant under conjugation – i.e. morphisms of $G$-sets.
A: To add to the other good answers here, there's a family of examples that could be seen as a bit trivial. But in a sense they give the simplest answer to your question.
Let $X$ be a partially ordered set, viewed as a category in the usual way: the objects of the category are the elements of $X$, and $\textrm{Hom}(x, y)$ has either $1$ or $0$ elements according to whether $x \leq y$ or not.  If $X$ is a meet-semilattice, i.e. any finite set of elements has a greatest lower bound, then the corresponding category has finite products.  It's cartesian closed if for any two elements $y$ and $z$ there's an element $y \to z$ with the property that for all $x \in X$,
$$
x \wedge y \leq z \iff x \leq y \to z.
$$
Here $\wedge$ denotes greatest lower bound.  So $y \to z$ is the internal hom $z^y$.  A poset with this property is more or less what's called a Heyting algebra.
I think this is an enlightening family of examples because in a poset, the external homs are pretty trivial (sets with at most one element), whereas the internal homs can be informative. For example, in a power set they give you the notion of complement (exercise!).
You can extend this family a bit. Again take an ordered set $X$, regarded as a category in the usual way. But now think about non-cartesian monoidal structures on it. For instance, take $X = \mathbb{R}$ with its usual ordering.  The cartesian monoidal structure is $\mathrm{min}$, but we could instead use $+$.  Then the internal hom is given by
$$
y \to z = z - y.
$$
Is that "unexpected"?  That depends on your intuition. But in this example the external hom only tells you the sign of $z - y$, whereas the internal hom tells you its actual value. In other words, it produces the operation of subtraction, which historically has proved quite important.
A: Here are a slightly different flavor of examples. Of course as mentioned by Peter Lumsdaine, the trick is always the same: looking at categories where $Hom(I,X)$ looks quite different from the "underlying sets of $X$"
Consider the category whose objects are sets, and whose morphisms are relations, i.e. subsets of $X \times Y$ (with the usual composition of relation).
It has a monoidal structure given by the product of sets. The exponential for this monoidal structure is simply the product, indeed,
A relation from $X$ to $Y \times Z$ is the same as a relation from $X \times Z$ to $Y$.
Other similar examples include:


*

*The 2-category of sets and spans of sets (with the products of sets as monoidal structure).

*The category of small categories and profunctors, with the products of categories as monoidal structure, here the exponential $[X,Y]$ is given by $Y \times X^{op}$

*The category of polynomial functors (which is actually cartesian closed) described in The cartesian closed bicategory of generalized species and structures by Fiore, Gambino, Hyland and Winskel. Where the exponential is a little more complicated, but still has the same flavor.
But one can somehow "explain the trick" behind them: One way to think about these example is that these categories are equivalent to categories where the Hom objects corresponds more naturally to a natural structure on the set of morphisms, but the equivalence of categories somehow makes the "underlying set" very different.
For example, the category of sets and relation can be seen as the category of "Free suplattices":
A suplattice is an ordered set with all supremums, and suplattice morphisms are order preserving map, preserving the supremums. Given two suplattices $X$ and $Y$ the set of morphisms from $X$ and $Y$ is naturally a suplattice for the pointwise ordering (induce by the order of $Y$), and this corresponds to a monoidal symmetric closed structure on the category of suplattice. So this is really a "non-example" of what is asked: the hom objects are really exactly the set of morphisms with a natural structure induced on them.
Now the forgetful functor from suplattice to sets has a left adjoint, sending any set to the suplattice $P(X)$ of subsets of $X$, sublattice morphisms from $P(X)$ to $P(Y)$ coincide with relation from $X$ and $Y$, and the monoidal structure described above is induced from the one on suplattices...
So our example become a non-example up to an equivalence of categories. Similar descriptions can be obtained for the other examples I have mentioned.
