Would you suspect that Apple would be OK with giving the app 3 minutes of background over and over, thus leaving the process in the background for say a hour?
No, Apple would not be OK with it, even if you could do it. They specified a 3 minute limit for a reason, to ensure that we don't have apps running in the background without the user's knowledge, consuming CPU cycles, memory, and draining the battery. (In fact, the "finite task" limit used to be 5 minutes, but several years ago Apple further restricted it to just 3 minutes.) Imagine a world where all app developers were routinely circumventing this 3 minute limit, our devices would have their batteries drained quickly and foreground apps would be less responsive and have less memory with which to operate.
Having said that, Apple has identified a very narrow set of operations that are acceptable to keep running in the background (VOIP, navigation apps, music apps, bluetooth operation, etc.), where the user has reasonable expectations about the battery and performance impact.
There are are also classes of tasks that employ some limited background capabilities (e.g. requesting time to complete some finite-length task, opportunistic periodic background fetch, significant change location services, giving the user a chance to respond to push or local notifications, etc.). The intent is to offer a meaningful balance between background capabilities while minimizing battery impact.
Bottom line, Apple otherwise discourages/curtails the use of indiscriminate background operation. In the App Store Guidelines, they explicitly say
2.16 Multitasking Apps may only use background services for their intended purposes: VoIP, audio playback, location, task completion, local notifications, etc.
...
4.5 Apps using background location services must provide a reason that clarifies the purpose of the use, using mechanisms described in the Human Interface Guidelines
Having said all of this, if you describe what precisely you need this background operation for, we might be able to describe which of the multitude of different background capabilities that Apple offers you could use to achieve the desired affect. All of these interfaces are designed to solve specific problems while balancing functionality with the scarce resources on our devices. But if it's something like "hey, I want to ping my server every five minutes", then no, Apple will frown upon that.
For more information, this is discussed in some detail in the Background Execution chapter of the App Programming Guide for iOS.