MrAl,What you could do is look at the "Nano" board of the Arduino line. It is smaller than one C cell.
You should also download the Arduino IDE and see if it looks like something you'd want to use.
If you want to use pure logic, we'd have to use a counter or two along with some gates. We'd also need a clock signal to get one counter to count maybe once every second to ding the bell once per second. Not sure what timing you want here though.
I am sorry about the delay in replying but I have been suffering from an infection. I have decided not to proceed with the Arduino approach as the learning curve seems steep. If I do decide to continue with this idea it will be using discrete components. Many thanks for the help. RMHC