What are some examples of research that does not solve a existing problem (maybe the problem is not obvious yet) but present a unique discovery?

Feynman noticing the periodicity of wobble and rotation of a dinner plate. The fractional quantum Hall effect. HTSC cuprates. Initial discovery of superconductivity. Michaelson Morely. Many discoveries of new plants and animals. Penicillin. Teflon. The discovery of the Americas.

The example that popped into my head was the discovery of the muon, about which I. I. Rabi famously quipped, "Who ordered that?" No one had predicted that any such particle would exist, and we still have no idea why it exists (more precisely, why there are three copies of all of the fundamental fermions). They were looking for pions, which happen to have a similar mass but are otherwise unrelated.

Since their discovery muons have seen a few practical applications, such as muon tomography.

It is a bit unclear what you mean or whether you are only interested in applications of research when you say "real-world".

Almost all research starts out without a well defined problem that is known in the literature. There are classic unsolved problems, of course, and people work on those, but graduate students are advised to avoid them. If a problem has been known for a while it has probably been worked on and the solution is (a) likely pretty hard and (b) the field hasn't yet developed the base to make a solution accessible. If hundreds of people have looked at a problem without a solution it is probably pretty hard.

The Four Color Theorem is instructive. Can you color every planar map with only four colors so that adjacent regions have different colors. The theorem has now been proven, but until computers came along the proof was pretty inaccessible. The proof is also unsatisfying to many mathematicians, since the analysis of a large set of cases that the proof requires looks pretty boring. Perhaps someone someday will have the insight for an elegant proof. But for a doctoral student to stake his/her future on finding that proof would be a tactical mistake.

Most research starts out with someone simply wondering about a question they have based on their readings. They don't find the question stated somewhere most likely, but they are puzzled by something and wonder if they can resolve their puzzlement somehow. "How do people balance themselves?" "How do bats find insects in the dark?" "Why do large dinosaurs have holes in their skulls?" "How does the local behavior of a real valued function affect the global behavior?"

A lot of research (doctoral level) starts out with an advisor giving a recent paper to a student with instructions to read it and then answer the question "What do you think about that?". Not very well formed. A research question might arise from their discussion. Is it correct? Can it be extended? Can it be combined with this other thing? Suppose look at what happens if we change this assumption?

The research grows out of those questions by finding some methodology that can get close to an answer.

But if you are only interested in applications, I think that the original researchers in most fields don't think much about that. Some do, of course, but more are interested in the pure knowledge that can arise.

The original researchers about bat echolocation weren't thinking about radar and how to catch speeding automobiles. That came later.

Most pure mathematicians study math for the ideas. My own dissertation was so esoteric that I assumed (not quite fifty years ago) that it would never find application in the real world. My results were unique, but not useful, most likely.. After thirty years or so, I was proven wrong when someone else found a "real world" use for what I did purely for the intellectual challenge (and the degree, of course). And I was surprised to see the application appear.

Applied mathematicians, on the other hand, do start with a given problem, but, again, it is probably a problem that they formulate themselves, rather than one that they found already fully formed in the literature.

Likewise, people in product development start with an idea for a useful product and work to create it. But that is a bit different from research as an intellectual activity. But no one looked in the literature for the problem "How do you create an iPod?" back in the day. The creation of the problem itself was an important part of the process. _