Now here is where I am confused. How many shields can be used with the Arduino?
shields are just stacking daughter boards, and the number doesn't matter, it is what pins they use, if two shields use the same pin they cannot be used at the same time.
you could just build a clock + keypad and have it turn on and off a FM radio via a relay, having the audio signals passing through the arduino is kind of wasteful anyway.
I wouldn't even bother using shields, a serial LCD just uses two pins, and using cables gives you more flexibility in design.