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:

Is $n$ a square?

Can $n$ be any natural number?

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 $S^{**} \cong n^2 S$.