Sequence of vectors spaces of linear transformations of vector spaces. Let $V_0$ be a vector space over some field k. Then the set of linear transformations $V_1 = \{T:V_0\rightarrow V_0\mid T\text{ is linear}\}$ is a vector space. 
Now, let $V_{n+1}= \{T:V_n\rightarrow V_n\mid T\text{ is linear}\}$. Does this sequence have any interesting properties? Is there a limiting vector space? For a finite dimensional $V_0$, I suspect that the limit will be a space with countably infinite dimension since $\dim V_n=(\dim V_0)^{2^n}$.
In general, in a closed category, does the sequence defined by $x_{n+1} = \text{Hom}(x_n,x_n)$ (where Hom is the internal Hom of C) for some $x_0$ have a limit or have some interesting properties? Is there a more algebraic definition for the limit if it exists?
 A: As I've already explained in the comments (and this here is again another comment, but too long), in order to get some interesting colimit, we need nontrivial transition maps $V_n \to V_{n+1}$. I don't see one for $n=0$, but there is one for $n>0$, and the colimit doesn't depend on the first step. So let's start with $V_1=\mathrm{End}_k(V)$, or more generally, let's start with any $k$-algebra $A$. Define $A_1 := A$ and $A_{n+1}:=\mathrm{End}_k(A_n)$ (endomorphisms as a vector space), this is again a $k$-algebra. Then the multiplication $A \otimes A \to A$ corresponds to a linear map $A \to \mathrm{End}_k(A)$, namely $a \mapsto (b \mapsto ab)$. This is even an injective homomorphism of $k$-algebras. For $A$ replaced by $A_n$ we get a transition map $A_n \hookrightarrow A_{n+1}$. We can consider the colimit $A_{\infty} = \mathrm{colim}_n ~ A_n$, which is a $k$-algebra. In most cases it is of infinite dimension. We have $k_{\infty}=k$, but I don't have any nice description $M_2(k)_{\infty}$. Note that that $A_n$ as well as the transition maps for $n>1$, thus also $A_{\infty}$, only depend on the vector space structure of $A$. So somehow this construction is quite strange. It is also not functorial with respect to homomorphisms, only with respect to isomorphisms.
