How many languages could you realistically learn?
How many languages is it possible to realistically learn and maintain fluency with? There's people that claim to have mastered dozens of languages, but this is obviously not realistic.
I believe there is not a single quotable number, but I know it is possible to properly learn a dozen languages. Watch Luca Lampariello on Youtube for an example.
However I do see what you mean: there are many people who like to "taste the surface" of languages, which is totally fine, but who also like to claim they can already speak them. So the real question is: when can anyone claim to have achieved a decent level of proficiency to make such claims? Only after answering that can you see the overall idea of what "realistically" would be.
Check out the book "Babel No More" ... it tries to answer this question. The big thing to remember is we all have a different idea of what "knowing" a language entails. It would appear there is no real limit to learning languages but that for most people with multiple languages there are "working languages" that they can recall and use with minimal effort, and "stored languages" that require refamiliarization, (maybe a day of reviewing the basics to get your mind in the right place) and any number of partial languages that are stored deeper and require more review to use. Then there's the level of "working" can you converse freely? Read? Write? Do you consider yourself fluent? Near Native? Are you great with grammar but your pronunciation is way lacking?
To reach genuine fluency, you have to live and work in a language for some time. I think that is probably the primary constraint - how many languages are you exposed to in a day-to-day setting, and how many times in your life you are able to change that day-to-day setting.
To be completely fluent, you do have to use a language regularly. I think that's the biggest challenge. I struggle to stay fluent in French because I never use it.
I think the other aspect that matters is how closely related are the new languages to the ones you already know. Related languages tend to follow the same grammar rules and use similar vocabulary. For instance, I can understand some Norwegian, Danish, Icelandic and Spanish from having studied Swedish and French, even though I'm not specifically studying those languages. Similarly I can understand bits of German and Romanian. Or when I'm listening to Russian or even Finnish, I'll pick up words here and there, because languages borrow words all the time (in contrast to the most common every day words that change the least).
English is a bit of an oddball, being a mash of Old English/Germanic and French.
To get an idea of what I mean by related languages, have a listen to Scots: https://youtu.be/cENbkHS3mnY
Scots is about as different from North American or Australian English as Danish, Norwegian, and Swedish are from each other. Some different vocabulary, some different pronunciation, and very similar grammar. With an open ear, you'll understand it. So you can see how learning one of the Scandinavian languages makes the others very easy to pick up, for instance.
I've met many people who are trilingual and quadralingual, and had a language teacher who was pentalingual, who had casually picked up English when he moved to Canada. Most of my colleagues speak two or three languages fluently.
I suppose the most fair level of "mastering" a language would be knowing a language to the extent of an uneducated native speaker. A great many people who study English as a second language know it better than many native speakers. I don't think it's fair to include slang or dialetic words, since they evolve quickly and vary from place to place.
Learning a language to that level takes somewhere between 500 and 2000 hours, depending on how similar the language is to one the student already knows. Maintaining a language takes a couple hours of usage a week (in which I would include thinking in that language as you go about your day).
So I think it's very reasonable for someone to be fluent in half a dozen languages and to know a dozen more to some extent. I think a person who has a knack for languages and who is passionate about languages could master a dozen and be conversational in far more over a couple of decades, especially if their profession involved using those languages and the languages are from the same families. Pope John Paul II spoke eight fluently and knew another four, for instance.