What is the name of $0.\overline{0}1$ Short question:
What is the name of the number closest but not equal to zero?
Long question:
Some programmers were discussing about the smallest number close to zero, which is Math.Epsilon or Math.MinValue depending on the programming language. It's value is limited by the definition of the IEEE float or double number format.
The discussion arose whether there is a similar number in Mathematics. We came up with the following idea of its value:
$$0.\overline{0}1$$
Is there a name of that number, the closest number to zero, not equal to zero?
 A: This number is known as an infinitesimal. In modern analysis it doesn't exist as part of the real numbers, but in the days of Newton and Leibniz it was used quite often. Note, however, that the concept of "the closest number to zero" is internally inconsistent, because then $\frac12x=x$, so $2x=x$, and that violates the rule that if $0<a$ then $a<a+a$.
In the mid-19th century it was shown that calculus can be based on finitary means which does not require infinitesimals, but in the mid-20th century it was shown that calculus can be constructed in such way where infinitesimals exist. However this will not add a lot of new theorems to calculus, because what is provable with infinitesimals will be provable without it; and it will cause the field of numbers that we get with infinitesimals not to be Dedekind-complete, which is a very nice property that we use often.
The field where infinitesimals are used is called non-standard analysis, and it uses fields of numbers called hyperreal numbers (but also others).
A: Besides what Patrick Da Silva already told you, you should understand that there is a difference between the limits of a programming language -the smallest (positive) number that it can handle-, due to physical limits of our computers, or of our software, and mathematical objects, such numbers, that don't have such limits.
For instance, according to Mathworks, the smallest positive number that Matlab can work with is
$$
2.2251e-308 = 2.2251 \times 10^{-308} \ ,
$$
which is pretty small, isn't it?
But your mind is more powerful than Matlab and has no problem to divide it by $2$, getting
$$
1.11255 \times 10^{-308} \ .
$$
So, yeah, you've approached zero better than Matlab can: isn't that amazing?
Ok, on a more serious tone, you could talk about something similar to the "smallest positive number" with non-standard analysis and its notion of infinitesimal number, but I'm not sure you would like it.
A: There is no such real number. This is simply because if you would have such a number, you could divide it by two and again get another number closer to your previous one, and still not zero.
A: There is the concept of "machine zero" which is useful. In many programming languages, if you start with a nonzero floating point value $x=1.0$ and repeatedly divide by (say) $2.0$, as in the assignment $x\leftarrow x/2.0$, you will eventually reach a point where the machine rounds the answer to zero. The value immediately before the division was rounded to $0$ can be regarded as the smallest positive number you can work with. This sort of thing is used to avoid accidentally using an "increment" or "step" value that doesn't really change a number when you add it to the number, so you don't fall into an endless loop when trying to step across a finite interval.
A: In terms of the extended decimal expansion of a hyperreal introduced by Lightstone, the number you wrote down makes sense as the infinitesimal with the first nonzero digit "1" at a suitable infinite hypernatural rank. You might want to think what $1-0.\overline{0}1$ would look like but this tends to provoke passions at SE :-)
As far as names are concerned, Leibniz referred to such numbers as being inassignable as opposed to usual numbers we could measure (the assignable ones).
