Certain real numbers can be approximated arbitrarily well by computable functions. If we introduce halting oracles, then more real numbers can be "computed", like Chaitin's constant or the limit of the Speckner sequence.

Using Turing machines for simplicity, we can define an order-$\alpha$ oracle Turing machine for a countable ordinal $\alpha$ to be a Turing machine that can access an oracle to solve the halting problem for any order-$\beta$ oracle Turing machine where $\beta<\alpha$. (This can be done by, say, taking a bijection between the integers and the ordinals less than $\alpha$ and using consecutive symbols on the tape to encode ordinals.)

Let's define the computability ordinal of a real number to be the smallest $\alpha$ such that there exists an order-$\alpha$ oracle Turing machine that approximates it infinitely well. Does there exist a (countable) computability ordinal for every real number?

This follows from the Axiom of Constructibility (related question V=L and a Well-Ordering of the Reals). I think it looks likely equivalent to AoCon, as if every real has a computability ordinal then we can use that and a numbering of the Turing machines to well-order the reals, but I don't know how to prove it.