I would caution you however to be careful with your programming. A given address within the EEPROM can only be written to 100,000 times. That's a whole lot of button pushes, if programmed correctly.
However, if programmed wrong, those 100,000 cycles can be burned up in a matter of minutes. Assuming you have a variable Holding the button press you just read (currentButton) and another holding the last button pressed (lastButton) then never write to EEPROM if currentButton = 0 (no button pressed) and also never write to EEPROM if currentButton = lastButton.
Take those precautions and all will be well for a long, long time.
If you want to write to EEPROM often (every second recording the temperature for example) then you should consider using a separate SD Card rather than the EEPROM on the ATMEGA (Arduino) chip. First off, assuming you write to a file, there is circuitry built into the SD Card to spread use around and secondly, you can replace the SD Card really easy and don't Need to reprogram the Arduino.