Paul lived in North Arabia (aka Nabatea) for almost three years - from about March, 34 AD, to perhaps August, 36 AD. In mid-50 AD, Paul mentioned this to the four churches in South Galatia. In late 56 AD, Paul said even more to the Corinthians.
"Five times I received from the Jews thirty-nine lashes. Three times I was beaten with rods. Once I was stoned... I have been on frequent journeys, in dangers from rivers, dangers from robbers, dangers from my countrymen, dangers from the Gentiles, dangers in the city, dangers in the wilderness..."
The stoning was outside Lystra. We don't know where the five lashings took place (or why Luke didn't mention any of them). One of the rod-beatings was in Philippi. But where were the other two beatings? Possibly in Arabia.
Also, the "dangers" from the Gentiles, in the city and in the wilderness might all refer, at least partly, to Paul's time spent in Arabia as well. Especially since Arabia was apparently on Paul's mind while composing this passage. A few lines later, he says, "In Damascus the ethnarch under Aretas the king [of the Nabateans] was guarding the city of the Damascenes in order to seize me..."
Any reconstruction of Paul's time in Arabia has to consider the beatings. Paul definitely got in trouble there - enough to get chased all the way to Damascus, over 270 miles away from Petra, the capital of Nabatea. It's very possible Paul was taken, beaten and released once before he got in trouble a second time. Or even a third.
I wish we had records of Nabatean punishment customs. If Paul was facing a "third strike", that would explain his flight back to Damascus. But that's just wild imagination - there's no good reason to imagine that as the most plausible situation. We'd need more basis for some reasonable speculation, at least.
Still, Arabia is a possibility* for one if not both of the extra beatings. Plus some of the "dangers".
But these are just considerations to keep in mind until I find another clue or until extra evidence presents itself.
*Illyricum is another possibility. Paphos, Cyprus, claims one of the beatings. And since we know Luke deliberately leaves out certain facts, these beatings could be in any city Paul visits during Acts! It's quite a problem. Some reasonable (not wild) speculation can be merely a beginning, not an end of the solution. But the problem should not be pronounced unsolvable. As long as there are a finite number of possibilities, we may be able to find some degree of reducability among the options. More on this another day, perhaps! ;)