Assuming that you impose no restrictions on the reduction, then any nontrivial problem $L$ in R is complete for the class. By nontrivial, I mean that language should contain at least one "yes" and at least one "no" instance. The reduction is very simple: Suppose $L'$ be any language in R.
We fix a canonical "yes" instance $x_1$ and a canonical "no" instance $x_0$ in $L$.
Since $L'$ is in R, it is decided by some algorithm. Solve the given instance using this algorithm.
If the result is "yes", output $x_1$; otherwise output $x_0$.
It is clear the above reduction works.
This situation has analogues in complexity theory as well: any nontrivial language in P is P-complete under polynomial time reductions. To overcome such arguably silly conclusions, while reducing a problem $A$ to another problem $B$, the usual understanding is that the reduction is allowed less resources than the algorithms solving either $A$ or $B$. For example, while logspace reductions make sense for P, polytime reductions do not; on the other hand, polytime reductions are useful while studying NP or PSPACE.