I'm an extreme mathematical layman, so please excuse the probable ignorance and awkward phrasing of this question.
Is there such thing as a kind of number which has an uncountably infinite amount of digits after the decimal point?
To give an example of what I mean, say such numbers exist and we'll call one of them S. The digits of S are such that every rnth place (and rn can be any positive real number) corresponds to the first decimal place of sin(rn).
The 0th place of of S is 0. What follows are infinitely many zeros until the digit place that corresponds to sin^-1(.1) is reached, and the digit of that place will obviously be 1. Long after that, the .5th place is 4, the .61999th place is 5, the 1st place is 8, the πth place is 0, the 6000π/e-th place is 7.
So, S will have a rather dull repeating pattern of 0.000...111...222...333..., eventually returning to 0s at the πth place and then cycling on again.
Another such number could be C, which has a similar rule but the rnth place corresponds to the first decimal of cos(rn) instead. C would look like 1.999...888...777, and so on.
I know S and C can't be real numbers (if these can even be considered numbers at all), because the cardinality of a real number's digits is equal to the set of the natural numbers. The cardinality of the digits of S and C is equal to the set of real numbers.
Now, is all this just useless mumbo-jumbo hiding behind ellipses? Supposing it's not, what would the cardinality of the set of all these weird numbers like S and C be? Would it be larger than the set of all real numbers?