I can't be bothered making yet another nodeshell for the smallest natural number. Just deal with it.
There are only a finite number of people who have ever existed on this earth who knew of numbers, they lived for a finite amount of time, and the process of considering a single number takes a non-zero amount of time. Therefore, a finite number of integers have been considered. Since there are a infinite number of integers, there are a set of numbers that have never been considered. What is the smallest number within this set that has never enlightened a human mind?
The problem is that the smallest natural number that has never been considered is a moving target. Consider the existance of such a number. At that moment, before we are able to place a value on it, we have considered the smallest integer never considered - a non sequitur if there ever was one. So can a smallest integer never considered exist? Maybe only whilst we don't think about it.
So should we say "the smallest integral value that has never been considered?" Let's suggest that 5 has never been considered before, and that we work that out. But now it has, once more, been considered. So what is the next number? 6? 7? 8? Some number much larger like 23? But we have now named in our thoughts these numbers. This does exist: it is just that finding its value removes its magic.
The smallest integer never considered has now changed.