Could you explain Perron's paradox to me, please? This is Perron's paradox:

Let $N$ be the largest integer. If $N > 1$, then $N^2 > N$, contradicting the definition of $N$. Hence $N = 1$.

What does it mean? I get from it that a very large number does not exist or $\infty=1$. Am I right? Or maybe the paradox is wrong?

It means that if there is a largest integer, then that integer is $1$.  That assumption is what led to a silly result.
The point of this "paradox" is to not assume that something exists; the question of whether it exists or not is important to investigate.  It has nothing to do with infinity.
