My question:
I have an iOS app, it is minimized, and I want it to start when user press home button three times (or, for example combination of buttons, like “HOME+Volume-” and so on). Can I make it for non-jailbroken devices, only with native iOS functions?
It’s not possible to customise iOS like that.
Without jailbraking, apps can only be opened by using custom URL schemes from other apps or by normally tapping on the on the home screen. To have an app open like this you would have to modify the OS and there is no way you can do that without a jailbreak.
Standard OS functionality like pressing the home button multiple times cannot be overridden because Apple is really against modifying any part of way the user interacts with apps or the OS.