Multitasking

You already admitted it, people want things like the mp3-player to run in the background. So just expand that: a streaming application, a downloading application, a networking application ... and yes, you can solve that in other, more abstract ways. But even if you provided apis to get such jobs done in the background (like the push-notification on the iphone), it is still limited to the functionality of the apis and not open for new ideas.

And on top of that, people do want fast application switching without loosing state in either of their applications. The state-saving and restoring is a solved problem, but on most devices it is still too slow to load the application from scratch and restore the state. So people ask for multitasking, i.e. holding more than one application in memory to switch seamless between them. Just to give an example, i want to write a mail to make an appointment, and have look something up on a website or in my calendar. Then i have to quickly chat with a friend/colleague to clear something up, and while still editing on the mail i start maps to see how much time i need to plan for travel. I never need more than one application, but i would absolutely hate if the browser needs to reload the website every time or if maps would have to reload the tiles every time or if the switch between them would take seconds.

Android does it the best way. There are services that run in the background for everything where you need to get some job done but do not need to look at it. There is multitasking for fast switching between applications and there is state-saving and restoring when there is not enough memory to hold all applications.