Metric number theory, diophantine approximation, and the probabilistic method provides many such examples.

Example 1: It is known that for almost all real numbers (all except a set of Lebesgue measure zero), the denominator of the $n$-th convergent $q_n$ satisfies $\lim_{n \rightarrow \infty} q_n^{1/n} = \exp(\pi^2/(12 \log 2))$. However, as noted in my question here, it seems difficult to produce many examples. Standard numbers such as $\pi$ or any algebraic number of degree larger than 2 are conjectured to have this property, but it cannot be proved.

Example 2: Any example generated by the (infinite) probabilistic method of Erdos. For instance, it is known that there exists a subset $B \subset \mathbb{N}$ with the property that every sufficiently large number $n$ may be written as a sum of at most 2 elements of $B$, and that the representation function $R(n)$ of the number of ways of writing $n$ as a sum of at most two elements of $B$ satisfies $R(n) = \Theta(\log n)$. To date, no explicit example of such a $B$ is known.

Example 3: Similar to example 2, it is known that a suitable random set satisfies the result of the Gauss circle problem, but it not known whether the squares themselves have that property.

Example 4: In my question asking for whether one can prove the existence or non-existence of curves of low degrees on algebraic varieties of higher dimension, the answer received is that it's doable in 'general' surfaces, but very hard to do for any specific variety.