The Internet tells me there is some work on languages that permit general recursion but carry information about possible divergence in the type system. For instance, the simply-typed language Koka and the dependently-typed language F-star appear to have a built-in primitive "divergence" effect whose presence is inferred by termination-checking.
From a logical perspective, general recursion leads to inconsistency by allowing the definition of non-terminating programs of type $\bot$, so unrestricted general recursion is unacceptable in a proof assistant. But marking general recursion by a divergence effect in the type system ought to restore logical consistency in the pure fragment. (Of course, in a dependent type theory one also has to decide to what extent types can depend on possibly nonterminating computations, and hence whether the latter may be executed during typechecking.)
Aside from explicit general recursion, non-termination and hence logical inconsistency can also be derived from universe inconsistencies such as $\rm Type:Type$. Has anyone ever considered a dependent type theory / programming language in which universe inconsistencies are permitted but similarly guarded by a built-in effect?