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.
You are confusing numbers with numerals. Numerals are symbols that represent numbers. Numbers do not have any intrinsic representation as sequences of digits or anything else. Instead, we devise different schemes for representing numbers with numerals. For example, in one scheme, we use sequences of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to represent certain numbers; the numeral 119 represents a certain number. But there is nothing privileged or special about this numeral; in a different, similar system, the same number is represented with the numeral 1110111; in a different, less similar system the same number is represented with the numeral 百十九, in another system it is represented with the numeral CXIX, in a different system it is represented with the numeral one hundred and nineteen, and in a different system again it is represented with a certain pattern of electron flow in a chunk of silicon.
So the question of whether a certain number "has digits in it" is a category error. Numbers never have digits. Some systems of numeration use digits, and numerals in those systems have digits in them. But the number of digits will depend on which system you are using. 119 is a three digit numeral, and 1110111 is a seven-digit numeral, but they both represent the same number.
The question that does make sense to ask is whether a certain system of numerals can represent a certain number. For example, some systems are able to represent the number one-half. One might write it in one system as $\frac12$, and in another system as 0.5. Some systems simply have no representation for one-half.
So we can ask if the standard decimal system, the one which uses digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, has a representation of the number infinity, and if so how many digits are used to represent it. And the answer is no, as usually understood, this system has no representation for the number infinity. (Or, more precisely, for any of the several numbers called "infinity".)
Best Answer
Your answer doesn't make much sense. It's equal to $1$ if anything (see Is it true that $0.999999999\ldots = 1$?).
You have to look at the definition: $m$ is the maximum if $m\in(0,1)$ and $\forall x\in(0,1),\ x\le m$. Clearly, this fails for any element $m\in(0,1)$ because $m<(m+1)/2<1$.