I don't have this bit (keyword...yet), but I think you may want to try the bluetooth low energy (BLE) bit. Here is the info: https://littlebits.cc/bits/bluetooth-low-energy
You can get that bit individually (http://littlebits.cc/bits/bluetooth-low-energy) or as part of the gizmos & gadgets kit (2nd edition only), although I don't think this particular kit has a button (it looks like it has a slide dimmer). Here is the kit info: http://littlebits.cc/kits/gizmos-and-gadgets-kit-2nd-edition
You may also be interested in the Arduino bit (http://littlebits.cc/bits/w6-arduino)or coding kit (http://littlebits.cc/kits/arduino-coding-kit). I am not sure if/how you can connect this to a phone, but you can certainly write a program to count the number of times a button is pressed and print to the little window that pops up (sorry, not actually programmer and I don't the technical name of the window). This kit will come with a button for your project too. In the Arduino IDE (https://www.arduino.cc/en/Main/Software) you will write C/C++ and it is not too hard to "connect" (not sure the technical term) to the processing IDE (https://processing.org/reference/) for more interesting projects (I think this is JAVA).
The other bit you may be interested in is the cloudbit. You can get this bit individually (http://littlebits.cc/bits/cloudbit) or in the smart home kit (http://littlebits.cc/kits/smart-home-kit) or the cloudbit starter kit (http://littlebits.cc/kits/cloudbit-starter-kit). It looks like both of these kits come with a button for your project. I think you will be able to find the information you need to get started with using the cloudbit here: http://littlebits.cc/cloudstart
You can definitely connect this bit to your phone and I assume you can use it to count a button being pressed, but I haven't played with this bit too much yet.