  We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a
computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to
$\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$.
We also call a set $A\subseteq\mathbb{N}$ regainingly approximable if it is
c.e. and the strongly left-computable number $2^{-A}$ is regainingly
approximable. We show that the set of regainingly approximable sets is neither
closed under union nor intersection and that every c.e. Turing degree contains
such a set. Furthermore, the regainingly approximable numbers lie properly
between the computable and the left-computable numbers and are not closed under
addition. While regainingly approximable numbers are easily seen to be i.o.
$K$-trivial, we construct such an $\alpha$ such that ${K(\alpha \restriction
n)>n}$ for infinitely many $n$. Similarly, there exist regainingly approximable
sets whose initial segment complexity infinitely often reaches the maximum
possible for c.e. sets. Finally, there is a uniform algorithm splitting regular
real numbers into two regainingly approximable numbers that are still regular.
