How do I get attention on a busy radio channel?
As Ron commented, the basic answer is that you try to anticipate when a short gap is going to be, and jump on before someone else does. It sounds a bit daunting, but it actually works out pretty well in practice.

That being said, it's not exactly first come first serve. ATC tends to prioritize talking to the aircraft which need the most immediate instructions. So, an aircraft on final is going to get more priority than one which hasn't even entered the airspace yet. Even if you manage to get your request in, if the controller has a more immediate issue to address, you may get completely ignored. Sometimes they'll come back with "calling approach, say again" or sometimes you just have to try again later.

Either way, it's usually not too difficult to stay out of trouble long enough to get someone's attention. If you're IFR or have already made initial contact, you can probably just continue on your path and you'll eventually become a big enough priority for them that they'll call you if you still haven't a chance to get a word in. If you're VFR and haven't made initial contact, you can probably just avoid the airspace for a few minutes.