Call an indecomposable module $M$ over a ring $A$ (restrict to finite dimensional algebras if you like or if it helps) $n$-almost reflexive in case $M^{\*\*} \cong nM$, when $(-)^{\*}=Hom\_A(-,A)$ and $nM$ is the direct sum of $M$ $n$ times for a natural number $n \geq 1$.


Questions:


1. Is $n$ a square?
2. Can $n$ be any natural number?
3. Can there be infinitely many different $n$ that can appear for a given ring?


Example: Take $A$ to be the quiver algebra with quiver having one point and $n$ loops and radical square zero. Then the simple $A$-module has $S^{\*\*} \cong n^2 S$.



Reflexive modules up to multiplicity
Consider the quiver algebra with quiver
$\require{AMScd}$
\begin{CD}
\bullet@>>>\bullet@>>>\bullet@<<<\bullet@<<<\bullet\\
@.@.@VVV@.@.\\
@.@.\bullet@.@.
\end{CD}
and radical square zero.


The simple module $S$ at the central vertex has $S^{\*\*}\cong 2S$, and by taking $n$ paths of length two into the central vertex, instead of two, any $n$ can be obtained.


For a finite dimensional algebra $A$ of dimension $d$, then at least for finite dimensional indecomposables it is clear that $n$ is bounded by $d^2$, since 
$$\dim\text{Hom}\_A(X,A)\leq \dim\text{Hom}\_k(X,A)=d\dim X$$
for any finite dimensional $A$-module $X$.


