@Joel David Hamkins I am sorry but I am still lost. I understand that there are superfast primality tests, but surely there is a limit. Just like there is a largest known prime with the current state of technology, it seems natural to ask if there a largest $N$ as in my question.

@Joel David Hamkins I think I may not have explained clearly enough what I meant. I do not understand how testing primality for any number can be feasible. Surely there will be numbers for which we cannot decide if they are primes or not, because they are too large. Otherwise what is the meaning of the announcement that $2^{82589933}−1$ is the largest known prime? I am thinking of an experiment, aided by a computer: take $N$, examine al numbers $1\leq n \leq N$ and for each decide if $n$ is prime or not. Surely there will be a largest $N$ for which this is possible, given a current computer.

I corrected the typo: $2^{82589933}−1$ is the largest known prime. A lengthy but hopefully clearer phrasing of my question: "What is the largest $N$ with the property that we can decide if $n$ is or isn't prime, for each $n\leq N$?"