I have an iPad mini (original model) running iOS 8.4 Recently I have noticed many apps seem to be too large for the screen; usually the top or bottom is partly off screen.
When I press the button, the home screen displays normally for a few seconds, then seems to "grow" e.g. the iPad in the top left now shows "Pad" wth the "i" off the screen.
This may have been since a software upgrade, but I can't be sure.
Perhaps you have somehow turned on the Zoom feature of Accessibility?
If you need more details, the following article might help
Check Settings->General->Accessibility->Zoom and made sure that Zoom is unchecked. Other than that, as far as I know, there is no way to change anything like that, at least without Jailbreaking (and I strongly recommend against that). It does seem like the kind of thing that iOS 9 might fix, though.
