I am wondering if there is some example of a famous or well-known mathematician who often had trouble with peer review, or who often had to publish in obscure journals because referees didn't 'get' what they were saying (there could be any reasons for these troubles).


Now, you might say this is paradoxical (a well-known mathematician is probably well-known because they typically don't have serious problems with referees), but there is a definite example in physics, in the shape of [Hannes Alfven](https://en.wikipedia.org/wiki/Hannes_Alfv%C3%A9n). Alfven was renowned especially for his work on plasma physics and was awarded the Nobel Prize for Physics, but even after winning the Nobel, he typically had problems with the peer review process (especially in the US), complaining that referees would automatically reject his papers because they could not understand the formalism he was using, forcing him to publish many of his articles in obscure journals.



Example of a mathematician who had problems with peer review system?
There does not seem to be a direct mathematical analogue of Alfven. Nobody who has won a Fields Medal or an Abel Prize has made well-publicized complaints about how they have had an unduly difficult time with the peer-review system.


Some partial analogues have been mentioned in the comments. Louis de Branges had trouble getting people to take his proof of the Bieberbach conjecture seriously, but there was a clear reason: de Branges had a well-deserved reputation for making repeated but incorrect claims to have solved famous open problems. Fourier's work met with resistance, but there were some legitimate objections that some crucial arguments were not fully clear; while the Mochizuki saga is still in progress, it seems that the clarity of certain crucial arguments is still in dispute. Galois also ran into difficulties, although [the story has been exaggerated](https://hsm.stackexchange.com/a/5343), and Galois was not a recognized figure at the time.


Why there is no direct analogue of Alfven is an interesting question. There could be a difference in culture. Mathematicians will often complain privately of getting an unfair rejection, but complaining too loudly or publicly about the lack of recognition of one's own work is generally regarded as whining, and is frowned upon. Mathematicians also take the general attitude that there are three primary reasons for rejection: (1) the work is wrong; (2) the writing is unclear; (3) the result is not regarded by community as sufficiently interesting. The first two are considered to be the fault of the author, and mathematicians tend to take a somewhat fatalistic attitude toward the third. Under these prevailing assumptions, it is hard to be viewed as taking the moral high ground when voicing a complaint about a rejection.


One example of someone making a public complaint was [Friedrich Wehrung](https://wehrungf.users.lmno.cnrs.fr/pubs.html) (search for the word "rejection"). His primary complaint, however, was that the entire *subfield* of lattice theory was being looked down upon unfairly. Similarly, you can find public discussion about how the peer-review system unfairly treats certain *groups* of people. But again, if you're looking for an *individual* analogue of Alfven, I don't think there exists a very good example.


