In a couple of web pages, I see that Legendre's constant is defined to be $\lim_{n \to \infty} (\pi(n) - (n/\log(n)))$ (for example, here and here).

Actually the first uses $\lim_{n \to \infty} (\log(n) - (n/\pi(n)))$, but I have the same question in either case.

To be honest, I just assumed that this was a typo for $\lim_{n \to \infty} (\pi(n) / (n/\log(n)))$ (that is, the prime number theorem). However, in both pages it looks to me as if the claim really is that $\lim_{n \to \infty} (\pi(n) - (n/\log(n)))$ exists and is $1$. Really?

I don't know whether to be more surprised that the limit exists or that its value is 1

Does anyone else find it surprising?