Classification of recursively enumerable sets
There are lots of these. Even with respect to Turing reducibility (which extends $m$-reducibility: if $A\not\ge_TB$ then $A\not\ge_mB$), the structure of c.e. sets is extremely rich.
That said, constructing an infinite non-recursive r.e. $A$ with $A&lt;_mK$ is significantly easier than constructing one with $A&lt;_T K$. The former was done by Post in 1944 who introduced the notions of creative and simple sets (each of which are necessarily non-recursive), proved that a simple set exists and that $K$ is creative, and proved that no creative set is $m$-reducibile to any simple set; the latter wasn't achieved until a decade later by Friedberg and Muchnik independently, and required a fundamentally new technique (the priority method).