The universe that Grothendieck intends to suggest by his notation is known in set theory as HF, the class of hereditarily finite sets, the sets that are finite and have all elements finite and elements-of-elements, and so on (the transitive closure should be finite). The set HF is the same as $V_\omega$ in the Levy hiearchy, and can be built by starting with the emptyset and iteratively computing the power set, collecting everything together that is produced at any finite stage. This is the smallest nonempty transitive set that is closed under power set. It satisfies all the Grothendieck universe axioms, except that it doesn't have any infinite elements, since none appear at any finite stage of this consrtruction.

There is an interesting presentation of this universe by a simple relation on the natural numbers. Namely, define $n\ E\ m$ if the $n^{\rm th}$ bit in the binary expansion of $m$ is $1$. The structure $\langle\mathbb{N},E\rangle$ is isomorphic to $\langle HF,{\in}\rangle$ by the map $\pi(n)=\{\pi(m)\,|\,m\,E\,n\}$, which set-theorists will recognize as the Mostowski collapse of $E$.

Since HF doesn't have any infinite elements, it is a rather impoverished universe for many applications of that concept. And so we naturally seek larger universes. But the difficulty is that we cannot prove they exist. The difficulty is not one of "discovery," but rather just that we can prove that the hypothesis of the existence of a univese containing infinite sets is too strong for us to prove from our usual axioms. The reason is, as has been remarked in some of the other answers and comments, all other Grothendieck universes have the form $H_\kappa$, the hereditarily size less than $\kappa$ sets, for an inaccessible cardinal $\kappa$. So this is just like HF, which is $H_\omega$, but on a higher level, and in this sense, these higher universes are not so mysterious. They are intensely studied in set theory, a part of the research effort in large cardinals.

In this MO answer, I mention a number of weaker universe concepts that we can prove exist, and which I believe serve most of the uses of the universe concept in category theory, if one wanted to care more about such set theoretic issues.

The universe that Grothendieck intends to suggest by his notation is known in set theory as HF, the class of hereditarily finite sets, the sets that are finite and have all elements finite and elements-of-elements, and so on (the transitive closure should be finite). The set HF is the same as $V_\omega$ in the Levy hiearchy, and can be built by starting with the emptyset and iteratively computing the power set, collecting everything together that is produced at any finite stage. This is the smallest nonempty transitive set that is closed under power set. It satisfies all the Grothendieck universe axioms, except that it doesn't have any infinite elements, since none appear at any finite stage of this consrtruction.

There is an interesting presentation of this universe by a simple relation on the natural numbers. Namely, define $n\ E\ m$ if the $n^{\rm th}$ bit in the binary expansion of $m$ is $1$. The structure $\langle\mathbb{N},E\rangle$ is isomorphic to $\langle HF,{\in}\rangle$ by the map $\pi(n)=\{\pi(m)\,|\,m\,E\,n\}$, which set-theorists will recognize as the Mostowski collapse of $E$.

Since HF doesn't have any infinite elements, it is a rather impoverished universe for many applications of that concept. And so we naturally seek larger universes. But the difficulty is that we cannot prove they exist. The difficulty is not one of "discovery," but rather just that we can prove that the hypothesis of the existence of a univese containing infinite sets is too strong for us to prove from our usual axioms. The reason is, as has been remarked in some of the other answers and comments, all other Grothendieck universes have the form $H_\kappa$, the hereditarily size less than $\kappa$ sets, for an inaccessible cardinal $\kappa$. So this is just like HF, which is $H_\omega$, but on a higher level, and in this sense, these higher universes are not so mysterious. They are intensely studied in set theory, a part of the research effort in large cardinals.

In this MO answer, I mention a number of weaker universe concepts that we can prove exist, and which I believe serve most of the uses of the universe concept in category theory, if one wanted to care more about such set theoretic issues.