Classification of monoidal closed structures The motivations for this question are somewhat vague probably. I am trying to have a better understanding of monoidal closed structures. In order to do so I would like to collect unusual examples and trivial patterns.

Q1.  Could you help me to generate a list of examples in which the monoidal closed structure is very different from the one related to Set? It happens very often that when $\mathsf{K}$ is cartesian closed and concrete $\mathsf{U}: \mathsf{K} \to \mathsf{Set}$, $\mathsf{U}(a^b)$ is a proper subobject of $\mathsf{U}(a)^{\mathsf{U}(b)}$. This is the kind of example that I want to avoid.
Q2. Assume that $\mathsf{K}$ is complete and cocomplete. Is it true that  $\mathsf{K}$ always admits a monoidal closed structure? Can you describe such a structure? If not, could you provide a counterexample?


This question is closely related or maybe even a duplicate of this other one.
Examples of interesting monoidal closed structures.

*

*Heyting algebras are precisely cartesian closed bounded lattices. In that case the internal exponentiation $a^b$ is the implication $b \Rightarrow a$.


*... I will add some if I think about them.
 A: Q1. Here's a large class of examples. In the linked MO question I described the cartesian closed structure on a presheaf category $[C^{op}, \text{Set}]$. This construction can be generalized to get non-cartesian structures as follows. If $V$ is a closed, symmetric monoidal, complete and cocomplete category (typical choices are $V = \text{Set}$ or $V = \text{Vect}$) and $C$ is a small monoidal $V$-enriched category, then the enriched presheaf category $[C^{op}, V]$ has a closed monoidal structure where the monoidal structure is given by Day convolution and the closed structure is given by a certain end. 
This is an absurdly general construction even in the case $V = \text{Set}$. If you want to work out an example, try taking $V = \text{Set}$ and $C$ to be the category of finite sets and bijections with monoidal structure given by disjoint union; then $[C^{op}, \text{Set}]$ is the category of combinatorial species with Day convolution giving the Cauchy product of species. 
For another class of examples see quantales. 
Q2. No. We will show that if $D$ is a noncommutative division ring, then the category $\text{Mod}(D)$ of right $D$-modules admits no closed monoidal structures. 
For starters, observe that in any monoidal category, the endomorphism monoid of the unit object is commutative by the Eckmann-Hilton argument. But the endomorphism monoids of right $D$-modules are (possibly infinite-dimensional) matrix rings over $D$, and hence are noncommutative except in the case of the zero module. Thus the zero module is the only possible unit object of any monoidal structure on $\text{Mod}(D)$. But: 

Lemma: In a closed monoidal category $C$, the unit object $I$ cannot also be the initial object unless $C$ is contractible (equivalent to a category with one object and one morphism). 

Proof. In a closed monoidal category, $(-) \otimes X$ has a right adjoint and hence preserves colimits, in particular the initial object. So if $I$ is the initial object and also the unit object, then we have both $I \otimes X \cong I$ and $I \otimes X \cong X$. Hence in this case every object is initial. $\Box$
So the zero module cannot be the unit object in a closed monoidal structure on $\text{Mod}(D)$, and hence there is no such structure. 
