A good rule of thumb is that anything over 10ms response appears laggy (not instantaneous) to a user. This is especially dangerous on touchscreen inputs, as they lack tactile feedback and therefore require the user to focus on the display until their input has been acknowledged.

I can't tell the specific concept you're talking about though. If it's the auto-repeat rate (your iPhone example), then of course there needs to be a delay before that kicks in, and a steady repeat rate. I'm talking about single button presses.

If you're talking about the requirement to debounce buttons, the proper way to do this is to act on the button press immediately, while setting a timer that disallows further button state changes while waiting for the contacts to settle. This time is a balance between how long it actually takes the contacts to settle, and how rapidly a user could possibly push a button (missing a deliberate button press is also extremely frustrating and dangerously demanding of user attention, even with tactile feedback).