Do multiple go-arounds on the same flight happen in commercial aviation?
None of four airlines I worked for (two commuters, two Part 121) had specific guidance as to the max number of go-arounds. However, as mentioned in the other answers, 2 or 3 was believed to be a good rule.

The only time I did multiple go-arounds, all missed approaches, was trying to get into Pullman, WA during a snowstorm in the late 1980s in an SA-227 Metroliner. I tried the VOR approach 3 times. Each time when we got down to the MDA, we did not have the forward visibility to descend further. Each of the 3 times the station personnel saw us pass over, and each time passengers looking out the windows saw the runway below us when we were overhead. That, of course, resulted in our being criticized when we arrived at our alternate by some passengers for not having landed at Pullman. I had explained en route to the alternate the problem with forward visibility, but when you've disrupted people's travel plans, they don't want to listen.

I briefly considered the feasibility, when we were over the runway, of entering a tight left turn to see if I could keep the runway in sight, but I rejected that. And, yes, I considered descending below the MDA before catching sight of the runway. That would have been doable as I was very familiar with the airport and knew that the gently rolling fields around the airport topped out well below the MDA.

The problem was that with all the snow on the ground and the snow in the air, we couldn't determine our altitude above the ground visually until we were actually over the runway. The fields around the airport had no trees, they were wheat fields in season. Even if I had chosen to bust the MDA, I still would have been too high for any kind of a normal landing when I caught sight of the runway end. If getting down was an emergency, we could have done it. It wasn't technically legal, but you could go into beta with the props and really drop out of the sky, then as you flared come out of beta. But it wasn't an emergency, so we went to our alternate, which was actually our next stop anyway, so all of the passengers weren't unhappy.

The advantage of multiple approaches is that conditions might get better while you're doing them, and you can get in. The disadvantage is that the nervous energy you're expending is telling.

Our SA-227s had no-autopilots, no radar altimeters, and, of course, no GPS. I imagine that these days that airport has one or more GPS approaches that would make landing in those conditions a piece of cake.