How about swiping left or right (or any gesture really) brings up a new number/page for that number? I'd hate to see the clean interface get changed by another button.
There are two types of "counter" that would fix my needs: a counter like (1, 2, 3.. and its datetime) and another one with just datetime info. With the former you can count the regular stuff like the number of times you do something and at what time. The latter can tell you at what time you do specific actions, like waking up.
This is a powerful thing as long as the user wants to look critically at the gathered data.
Unfortunately I don't have an iPhone or any of the i-family gadgets.
I think swiping would fit better though, although there could be a lot of swiping happening if you've got a few counters.
Also, editable titles would let you name your counter.