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
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
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".)