In our day and age you don't need to go to university or college to become an expert at something: encyclopedias, instant access to tens of thousands of textbooks and classics (both free and not), papers, discussion boards, educational videos and actual filmed lectures at the most prestigious of universities, Q&A sites where it's not rare for an answerer to have a PhD, the following question seems very topical (to me, at least).
What expression is best suitable for a person who does not hold a degree in something, but is still an expert in this something (sometimes ironically being a better expert than one who does hold a degree)?
The best expression that comes to mind is 'enthusiast', but it's not serious enough, in my opinion.
Best Answer
Enlightened people care about expertise in a particular area X, not about credentials, e.g. advanced degrees, professional certifications. The latter are meaningless compared to what a person knows or can do.
That said, the word you are looking for may be authority.
From Merriam-Webster:
An authority is a person who simply knows more about a particular something than anyone else, or most anyone else. Whether such a person has a degree or not simply shouldn't matter.