So, it's not a far stretch to take an arduino shield and turn it into a full on arduino. Really, an arduino is a development board for the ATmega328p. It has a USB to Serial converter (which I didn't put on my board), a power regulator, a crystal oscillator, a reset button, a power plug, and, of course, the ATmega328p microprocessor.
So, I added those to my board, and presto! You no longer need to add an arduino. It's there.
This does come with some trade offs. The first is that the board I made does not have a USB to Serial converter. This means you can't plug it into USB on your computer to program it. You will either need an FTDI converter (they sell them on Adafruit, Sparkfun, etc.) or, you will need an actual arduino to (see this tutorial on how to use an arduino board to program the ATmega328p chip).
Second, I used a different power regulator. This one can output twice as much current as the arduino one. So, it should have an easier time providing power to all the components.
Third, I used a resonator instead of a crystal oscillator. Resonators can have less accuracy than crystal oscillators, so the timing of my arduino may not be exactly 16 MHz.
But, it works! I've sent some boards out to Seeed Studio to get manufactured. They are a Chinese PCB fabrication company. The headband and arduino shield were sent to OSH Park. Seed Studio was cheaper for the boards that I'm making, but OSH Park makes nicer boards (OSH Park, by default, uses a higher quality process that I didn't pay for at Seeed Studio. I could have, to compare, but I didn't want to pay more).
Once I get the boards in, we'll see if everything works. The design is in Upverter, and can be found here.