If any recursively enumerable language can be reduced by a mapping reduction to a language $L$, then $L$ is called $RE$ complete. In that case, $L$ must be in $RE\setminus R$. But are there languages in $RE\setminus R$ which are not $RE$ complete? Can anyone give an example of such a language? I'm sure this is well known, but it's not well known to me, and google doesn't help.
Examples of such languages are not easy to describe, and I think no "naturally-occurring" example is known. However, Muchnik and Friedberg found examples in 1957, and Friedberg's example is here.