The set of real numbers are (usually) defined in a way that has nothing to do with decimal representations -- they are defined by their arithmetic and geometric properties. e.g. among other things, if $a$ and $b$ are distinct real numbers, then $(a+b)/2$ is a real number that is between them.
The set of decimals are defined as being sequences of digits: there is one place for every integer. e.g. $0$ corresponds to the one's place, $1$ corresponds to the ten's place, $2$ to the hundred's place, $-1$ to the tenth's place, and so forth. Each place gets a single digit (0 through 9) assigned to it. When we write a decimal like
123.45
we implicitly mean that all of the remaining positions get filled with zeroes. i.e. in the above numeral, the thousands place contains a zero.
The key point is that each place corresponds to an integer: there aren't any other places. If we write $n.\overline{0}$, meaning that the $0$ to the right of the decimal place should be repeated infinitely, this means that we have written a $0$ in every place corresponding to a negative integer. There aren't any places remaining to the right of the decimal place to insert a $1$! So the notation $n.\overline{0}1$ makes no sense if we try to interpret it as expressing a decimal number.
We could define other sorts of radix notation that extend decimals to have additional places to the right of the decimal place, but then we have to figure out what to do with such things.
The ordinary decimals are useful because we have a way to interpret any decimal number that only has finitely many nonzero digits to the left of the decimal place as a real number. And we also have rules for doing arithmetic with them. There are some ambiguities -- e.g. does $1.\overline{0} + 0.\overline{9}$ add up to $1.\overline{9}$ because there are no carries? Or does it add up to $2.\overline{0}$ because there is a carry in every place to the right of the decimal point? -- but these ambiguities are okay because we are interpreting both possibilities as being the same real number.
But if we extend the decimals, we no longer have the ability to relate them to real numbers. And if we want to do arithmetic with such things, we're going to have to do a lot of work to define the arithmetic operations and figure out if they have any of the familiar algebraic properties we're used to and so forth.
We can construct algebras in this way in which every number has a "next" number, but such things are going to have very little to do with real numbers.
The short answer is that you can't. But you can define any positive value which is arbitrarily small, in other words as small as you want it to be.
This means that just like how there is no "largest number" (since for any number $N$, we know that $N + 1$ is larger), there is no smallest positive number, since for any real number $N \in \mathbb{R}$,
$$\frac{1}{N} > \frac{1}{N+1} $$
Best Answer
It depends on the number system you're using.
If you're using the real numbers or the complex numbers, then zero has no reciprocal. In other words, $1/0$ is an undefined expression. Also, in these systems, there's no such number as infinity. In other words, $\infty$ is an undefined symbol.
If you're using the projectively extended real line or the Riemann sphere, then the reciprocal of zero is infinity, and the reciprocal of infinity is zero. In other words, $1/0 = \infty$ and $1 / \infty = 0$. (Note that the reciprocal of infinity is exactly zero, not infinitesimal. None of these four number systems contain any infinitesimal numbers.)
Out of these four number systems, the first two (the real numbers and the complex numbers) are much more commonly used than the last two (the projectively extended real line and the Riemann sphere). So much so, in fact, that we usually say "division by $0$ is undefined" and "infinity is not a number" without clarifying which system we're using.
The reason that the first two systems are more commonly used is that these two systems are fields, and the other two are not.