The 3 song rule in concert photography has been widely in place since the 1980's. Most photographers you'll run into have learned to capture what they can in those 3 songs, without flash. It's a rule that will become one of your (almost) constants when photographing bands.
So why does it exist? Is it really that much of a hindrance? How am I supposed to make the band look good in only three songs?
Well, the story is that back in the 80's, artists playing in New York (specifically Bruce Springsteen) were being bombarded by photographers (more like paparazzi) firing off their flashes into the artists' faces all night. So one night, once Bruce was off stage, declared that something needed to be done about it. Thus, the 3 song rule was born. This was also a time where MTV was a major thing, and artists wanted to look as good as they did in their videos for those photos as well.
In my opinion, this rule really isn't that difficult to work within if you're shooting as media. You're there to capture photos of the artist, and they essentially give you 15 minutes of time to do it. I don't know about you, but in 15 minutes I can pop up plenty of shots, and winning ones at that. I think the rule is a smart rule to have, and it's a respect thing as well to the paying audience members who have to tolerate us.
Now, the typical exception to this rule is when you're photographing for the band. The rule goes out of the window at that point generally. Also in today's times you'll find artists who will only have you shoot the first 2 songs, the first song only, the last 3 songs, songs 4, 5, 6, and any other combination you can think of. Today's era of concert photography is a big bucket of messiness at times, but now you know a bit more.