A definition of computable numbers that requires to "wait an infinite amount of time" to get the correct result; how to make this precise
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 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:


(Calculation fo approximations) There is a Turing machine $M$ which on input $n$ terminates outputs a pair of integers $(a, b)$ such that $|x - a/b| &lt; 2^{-n}$.
(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.
(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, $p &lt; x &lt; q$.


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. One can also study subclasses of computable reals, see again X. Zheng's work.
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 $0 \leq k &lt; \infty$. 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 $M$ 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 $M$ 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.