Not using a regulator

what happens if I was to run a project on batteries without a regulator. Say 3 x AAs for a micro rated to run from 2.5 - 5.5volts

A microcontroller, a digital pot and a few LEDs

What sort of problems occur, or what is the disadvantage?

Arduino Pro Micro?
Must be 8Mhz to run on 2.5-5.5volt.
Only LEDs with a Vf of <2.5volt can be used.
Most digital pots can only work with signals within the power supply limits.
What is this pot controlling.

The CPU clock rate usually has to be lowered to 8 MHz or less, if the voltage drops below 3V. Operation at 16 MHz is not guaranteed below that voltage.

NIck Gammon has an excellent tutorial on low power operation with a bare bones Arduino here.