Which is a technical way of saying "in real life people use finite rational or algebraic approximations for reals, so uncountability of reals and infinite precision aren't a problem".
Consider the following real number made of binary digits:
Enumerate all Turing machines and all possible inputs, iff the i-th machine/input combination holds, the i-th binary digit in our number is 0, otherwise 1.
This number is well-defined (once you fix your enumeration scheme).
But there's no finite algorithm to produce approximations in your sense.
What I was after were what's also called Computable numbers (https://en.wikipedia.org/wiki/Computable_number). But I used the more general term of co-recursion, that also applies to arbitrary other data-structures like infinite lists, or with some generalization, infinite event-loops where the important condition is that each run through the body of the loop only takes finite time.
People are just bit sloppy, and say algorithm when they mean something slightly different.
See https://stackoverflow.com/questions/28841260/what-is-the-dif... and https://en.wikipedia.org/wiki/Corecursion
Basically, you don't want an 'algorithm' here to produce the whole number.
All you need is some scheme that will produce the next digit in finite time (and the next one and the next one etc).