I've heard it said that Paul wrote other letters that didn't make it into the Bible, including at least one more letter to the church at Corinth, and a letter to the church at Laodicea. What is the source for these claims, and what might have happened to these letters?
Paul certainly wrote other letters, but they were either lost or were not theological. For example, 1 Corinthians 16:3:
Then, when I arrive, I will give letters of introduction to the men you approve and send them with your gift to Jerusalem.
Regarding one more letter to Corinth, that is the implication in 1 Corinthians 5:9 when he refers to an earlier letter:
I wrote to you in my letter not to associate with sexually immoral people.
Finally, it is worth noting that all genuine letters by Paul could be identified by people at the time as he ended them in his own writing (rather than dictating). 2 Thessalonians 3:17 says:
I, Paul, write this greeting in my own hand, which is the distinguishing mark in all my letters. This is how I write.
According to John MacArthur in his sermon titled Saved or Self-Deceived (available on YouTube here), he says Paul wrote two more letters to the church of Corinth (about 19 minutes into the sermon).
He explicitly says that they were not included in the Bible. "..a church to which by the time he writes 2nd Corinthians he's already written three other letters, 1st Corinthians and two other letters that aren't in Scripture." John Mac Arthur. As one of the most authentic bible scholars of our time, I would take his word for it.
Jerome Murphy-Oconnor says:
The intensity of Paul's relationship with the Corinthians is illustrated by the fact that he wrote more letters to them than to any other church. The New Testament contains only two letters, but these mention two others, the Previous Letter (1 Cor. 5: 9) and the Painful Letter (2 Cor. 2: 4). Hence, four in all.
From the end of the eighteenth century, however, doubts have been raised regarding the integrity of both i and 2 Corinthians. 2 The division of 2 Corinthians into two originally independent letters was postulated in 1776. It took a hundred years for the integrity of 1 Corinthians to be called into question. From that moment hypotheses became ever more complex as fragments from one letter were associated with those from another. This trend in New Testament research reached its climax with the thesis that originally the Corinthian correspondence consisted of nine distinct letters.
Jerome Murphy-O’Connor, Paul: A Critical Life, 1st edn (New York: Oxford University Press, 1998).p.252
See also Schmithals (1973). For a survey of the various modern partition theories, see Sellin (1987), 2965-8.
The missing reference to the letter to Laodicea is found in Col 4:16
Col 4:16 And when this letter has been read among you, have it also read in the church of the Laodiceans; and see that you also read the letter from Laodicea. (ESV)
where "from" actually should be translated as "of", meaning "the one I sent there".
The Apostle Paul spent many years in Rome, most of them as a prisoner of the Roman government. The evidence is that Paul arrived and was imprisoned in early 61 AD and liberated in 63 AD and that he was imprisoned again upon his return to Rome, possibly after returning from Spain. The thing that is noteworthy here is the amount of time he spent in captivity in Rome, most likely under a form of house arrest which allowed him to have regular visits from his contemporaries. No one will ever be able to say for sure how many letters he wrote from Rome to the Saints in Asia and the Roman Empire, but we can be sure this is how he spent most of his time while imprisoned, writing correspondence to the churches he had established throughout the Roman Empire.
As noted by others above, Paul himself speaks of other letters that had been written to other churches and individuals. It is important to understand here that Paul did not write his epistles or letters with the knowledge that they would one day be canonized and become a part of something we call the New Testament. These were letters and correspondence written to give clarification to unanswered questions and to inspire and uplift the specified recipients. These letters, we know, were circulated among the brethren because of a statement made by the Apostle Peter in 2 Peter 3:15 when he speaks of Paul's many letters. Peter here also speaks of how difficult these letters were to understand by the average reader.
At any rate, the fact remains that Paul surely had written many more epistles than appear in the current New Testament canon. The Roman Catholic Church is known to possess thousands of books and partial writings in the Vatican's library vaults. These books and parchments have been moved several times throughout history and the History Channel has a documentary showing the rarely seen writings. Within these writings there is undisputed truth of additional writings of the Apostle Paul. The big question now is what do these text contain? Would they give us any different insight into Christianity as we know it today?
Also consider this, if Peter speaks of Paul's writings in 2 Peter, and Peter was in Rome when he wrote this, most likely after Paul's death, then it would be very probable that some of Paul's writings were indeed being rewritten and collected by early Christians in Rome. I do not believe they were considered as equal to Scripture of the Old Testament, as we have no evidence of a Canon any earlier than about 130 AD.
I know you like a lot of references when post are made to this sight but this is given as a reference to anyone who wishes to investigate further.
Most modern scholars will point to some letters that were not written by Paul contributed to him and the other respondents above have pointed out there were letters that were not included. It is very possible there are Vatican controlled archives that contain letters attributed to Paul that were probably too late to have been of his authorship like Timothy letters. These pseudo Paul letters likely supported ideas that the canonical fathers did not want circulating about anymore. There is a real possibility that there are somewhere extant letters Paul dictated that would be intriguing to find, but I doubt they would really do nothing more than give a little insight as to some specifics of what was happening in the early churches or Paul's evolution? of thought and faith.