Is there a property in $\mathbb{N}$ that we know some number must satisfy but don't know which one?

For the first question, we know there is a smallest natural number $n$ such that $\pi(n)>\text{li}(n)$ where $\pi(x)$ is the prime counting function and li$(x)$ is the logarithmic integral, but we don't know what the number is.

EDIT: For the sake of having a more complete answer, I will include our discussion in the comments.

There has been much progress in determining an upper bound for the first integer at which this switch occurs, but still the number eludes us. Strangely enough, we know that there are infinitely many numbers for which this inequality holds.

Here's an interesting example that is the subject of recent (and celebrated) research:

Consider the set $\{p_1, p_2, p_3, \ldots\}$ of prime numbers, listed in increasing order. The $n$th prime gap is the difference $p_{n + 1} - p_n$. In 2013, Yitang Zhang showed that there are infinitely many prime gaps of size $\leq 7 \cdot 10^7$. Since there are only finitely many positive integers smaller than this size, at least one of them must occur infinitely often, but we don't know which. Since then, the Polymath project has improved that upper bound to $246$ (as of last December). The famous and longstanding Twin Prime Conjecture says that $2$ occurs infinitely often.

It is known that one of ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational, but no one knows which one:

W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. doi:10.1070/RM2001v056n04ABEH000427.