Simply speaking, Arduino is nothing more than IDE and bootloader flashed in to ATmega microcontroller. And you can see lots of Arduino projects built on custom PCB’s. It’s a matter of choice which way to go. The original idea of Arduino boards is stackability. This means if you need some functionality, just plug a shield board on top and you are ready. This is great. But probably no one wouldn’t mind to have a little prototyping area on Arduino itself, or maybe have less function for smaller price. Each situation is unique and gladly there are lots of choices or Arduino forms and sizes.
If you decided not to buy Arduino but build instead, then you can try Nanino – which is specially designed to be simple and doable with any home PCB technology. Its a single sided PCB with couple jumper wires. It has a small prototyping area where will be able to attach few electronics parts like temperature sensor or transistor switch. There is a connector to FTDI serial adapter. Form factor is kept to match Arduino shields. It has no power regulators, so be sure to feed regulated voltage when powering.