n-approximable functions
This is a special case of what is called an $f$-c.e. set. A set $A$ is $f$-c.e. iff there is computable function $g$ such that 


$\chi_A(x) = \lim_i g(x,i)$, and 
$\forall x \ |\{s \mid g(x,i) \neq g(x,i+1) \}| \leq f(x)$. 


If we consider $f(x)=n$ then we get the definition of $n$-c.e sets.

They don't need to be c.e. because c.e. is not closed under complement. Take two c.e. sets and take their difference. It is a $2$-c.e. set but it does not need to be a c.e. set.

Barry Cooper and colleges have been interested in these sets and the notion of computability in limit. You may want to check his "Definability in the Real Universe" article from "Computability In Context: Computation and Logic in the Real World".