Although you're talking about navigating away from something, and not to it, navigating away from something is also a navigation goal in itself. So I present you the infamous Three clicks 'rule' which for a long time was the accepted wisdom on how many clicks (or taps in your case) was acceptable for a user. However as people have learnt more about usability, and studied it with real scientific experiments, we have the three click rule myth.
What's important is not how many clicks/taps it takes to do something, but what the user thinks/feels about those clicks (the information they have). For example, I'd much rather click four times to get somewhere if each time I clicked on a meaningful label which confirmed to me my end goal would be validated, whereas if I was just randomly exploring then even clicking twice might feel like a waste of time.
This "how I feel it performs, is more important than how it actually performs" attitude has become the new norm. It's the principle behind usability in Googles new SPDY protocol.
In your case: Three clicks is okay, but to be safe, give some explanatory text. For example instead of 'back' have 'back to [last page name]', so they can identify with that navigation process and see where it's going. If you need more than three or four levels, Consider adding a separate smaller button beside your existing back button which 'roots' the user back up to the base.