At least 3.
0th Corinthians (yes)
Based on the chronology provided in Acts and Paul's other letters we can date 1 Corinthians to approximately AD 54-55, while Paul was in Ephesus, and we can date 2 Corinthians to AD 55-56, shortly after Paul left Ephesus. It won’t work to try to make 2 Corinthians chronologically the “first” letter (as some have suggested swapping 1 & 2 Thessalonians or 1 & 2 Peter chronologically). So since 1 Corinthians tells us that there was another letter prior (see 1 Cor 5:9 as the OP has noted), that gives us a minimum of 3 letters.
As early as the writing of 1 Clement (usually dated to AD 96, but see a thoughtful argument for AD 70 here), this earlier letter was apparently out of circulation. Clement refers to Paul’s 1st letter to the Corinthians (implying he knew of more than 1), but quotes the text we know as 1 Corinthians, meaning the prior letter (can we call it 0th Corinthians??) was probably not known to Clement. (see 1 Clement 47: 1-3)
Was there a fourth? (probably)
The “severe” letter is a pretty strong candidate for a fourth letter to the Corinthians (not chronologically 4th though). However, is it possible that this “severe letter” is actually a reference to 0th or 1st Corinthians? 2 Cor 2: 1-4 would be a very odd description of 1 Corinthians. Some have said it’s possible. Maybe. I think it extremely unlikely. It sounds like the "severe" letter was written in some haste, and Paul shortly followed it with 2nd Corinthians to smooth things over. 1 Corinthians was clearly not written hastily.
Is it possible that the “severe letter” is a reference to 0th Corinthians? We don’t know much about 0th Corinthians, except the brief mention in 1 Cor 5 that indicates Paul told them to be mindful of the company they spend time with. Could Paul have come across stronger than he intended? I’d have to grant it’s at least possible.
So I suppose it is possible that the “severe letter” is 0th Corinthians, but it does not strike me as the most intuitive reading. Paul has already written a very lengthy epistle in between, and if he felt bad about how he communicated in 0th Corinthians, 1 Corinthians offered him plenty of opportunity to set the record straight. So I’ll grant that it’s possible that 0th Corinthians and the “severe” letter are one and the same, but I lean towards the view that the “severe” letter is its own epistle. I’ll call it Corinthians 1.5. That gets us to 4 epistles.
Was there a fifth? (if so, it has less evidence than Atlantis)
The only case I see for getting 5 Corinthian letters is to allow that all four letters above are distinct letters and additionally, 2 Corinthians is a composite. Making Pauline letters into composites is popular in some camps (Romans 16 anyone?), although I cannot help but believe that it’s generally an effort for people to read their own presuppositions into the text. If they don’t want Paul to have said something, or a particular statement at a particular time doesn’t jive with a personal-favorite chronology, we’ll eliminate the problem by calling it a variant. A very big variant.
I see a real problem with this approach. Textual criticism identifies variants based on differences found in the text in the manuscript tradition. Identifying a variant for which there is no text in the manuscript tradition is a tricky business prone to speculation.
Since there is exactly 0 manuscript evidence (and 0 patristic evidence too!) in favor of the view that 2 Corinthians is a composite of two or more letters, I’ll stick with the evidence and conclude it really is just one letter.
Why are some of these letters missing?
In Paul’s day, Google didn’t automatically create a sent folder archiving your correspondence [citation needed]. It was common for a writer to make their own copy of the letter to keep on file (see Reece, Steve. Paul's Large Letters: Pauline Subscriptions in the Light of Ancient Epistolary Conventions). Reasons this would be helpful include:
They could resend if the original was lost
They could reference it in reading a response
They could quote it later if they liked the way they said something
They could call out a forgery sent in their name
E Randolph Richards has considered how the Pauline letters came to be compiled as a set, and he pointed out the extreme unlikelihood of someone traveling all around the empire years later prospecting for letters of Paul in order to compile them. An overwhelmingly simpler conclusion is that Paul (or those with him) kept a “sent folder” of some of Paul’s letters. (I cannot for the life of me track down his article I read on this but I'll be happy to add a link if I find it or someone else does)
This may, in fact, be what Paul is requesting in 2 Timothy 4:13:
Bring the cloak that I left with Carpus at Troas when you come—and the
books, especially the parchments.
If this is true, then the missing Pauline letters can perhaps best be explained by the hypothesis that not everything Paul wrote had a “sent folder” copy made at the time (especially if it was sent hastily?), and so when Paul’s letters were compiled, letters like 0th Corinthians and Corinthians 1.5 were left out.
Conclusion
I suspect the compiler of the Pauline letters--perhaps one of his close colleagues in Rome after Paul's death--didn’t have a copy of 0th Corinthians or Corinthians 1.5 on hand, and they were lost to history because they didn’t make it into this Roman set of Pauline letters. Clement’s reference to 1 Corinthians as Paul’s first letter supports this conclusion. (Clement already knew about a set of Pauline letters but didn't know about 0th Corinthians)
The surviving evidence of Paul’s correspondence with the Corinthians makes a pretty solid case he wrote them at least 3 letters, and a decent case that he wrote 4.
