What is the maximum number of LEDs you can use with your lilypad?

I would like to have 1 lilypad LED per pin and I am using the ATMEGA328 Lilypad Arduino with a 5V power source. How many LEDs in this formation can I have without frying anything?

Depends how much current each one draws, and how much current you draw altogether. Individually, limit it to 20mA /pin. In total, limit it to 200mA. So 10 LEDs at 20mA, or 20 LED at 10mA or less.