I would argue that infinite-precision numbers don't "exist", in a colloquial sense at the least. The only sensible way to talk about real numbers (not the set, mind you... lol) in my opinion is to define the precision. So sqrt(2) = 1 to one significant digit, 1.4 to 2 significant digits, etc.
If you define numbers this way, then certain irrational numbers - and all rational numbers - exist.
So, to answer your question, no. I don't think that any numbers "exist" as a limiting process of algorithms. I believe numbers exist which are the output of some algorithm which computes them. Non-terminating algorithms don't produce any numbers.