This is not a research-level question, but since the general level of interest seems high, here is an answer. I cannot guess from your question whether you're shooting for something that will result in the usual computable numbers, or you're trying to surpass that.

First we have [Turing's definition](https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf) of **computable real number**, and it is the one others have been commenting on: a real number $x \in \mathbb{R}$ is said to be computable if any one of the following holds:

1. *(Calculation fo approximations)* There is a Turing machine $M$ which on input $n$ terminates outputs a pair of integers $(a, b)$ such that $|x - a/b| < 2^{-n}$.
2. *(Calculation of digits)* There exists a Turing machine $M$ which runs forever and writes out the digits of $x$ on an infinite *write-once* tape. That is, once it writes a digit, it cannot change it.
3. *(Calculation of neighborhoods)* There exists a Turing machine $M$ which on input $(p,q)$, where $p$ and $q$ are rational numbers, terminates if, and only if, $o < x < q$.

There are many other equivalent definitions.

We can also ask about various other kinds of computability, and we shall discover a hierachy of classes of reals, see for instance X. Zheng's [Classification of the Computable Approximations by Divergence Boundings](http://www.sciencedirect.com/science/article/pii/S1571066107000199?via%3Dihub). One can also study *subclasses* of computable reals, see again [X. Zheng's work](http://link.springer.com/chapter/10.1007%2F11780342_60).
For instance, we can try these:

* *(Mind-change computability)* A real $x$ is **computable with $k$ mind changes** if there exists a Turing machine $M$ which runs forever and writes its digits on an output tape. While so doing, it may change its mind about any particular digits at most $k$ times, for some fixed $0 \leq k < \infty$. A variant allows computations where every digit eventually stabilizes, i.e, the number $k$ is not fixed to be the same for all digits.
* *(Oracle computation)* A real $x$ is **computable with oracle $A$** if there is an [oracle Turing machine][1] $M$ such that $M^A$ computes $x$ (in any of the senses above).
* *(Infinite-time Turing machine computation)* A real $x$ is **infinite-time Turing computable** if there exists an [infinite-time Turing machine](https://arxiv.org/abs/math/9808093) $M$ which computes $x$ (in any of the senses above).
* *(Definable real number)* A real $x$ is **definable**, say in the language of set theory, if there exists a formula $\phi$ such that $\phi(x)$ holds and, if $\phi(y)$ holds for any $y \in \mathbb{R}$ then $y = x$.

I am guessing you had in mind some sort of mind-change computability. It is stronger than the usual computability.


  [1]: https://en.wikipedia.org/wiki/Oracle_machine