Maintaining time in Arduino
I see from eBay that you can get the DS1307 for around 10 cents. Is that really too much to pay? You also need a crystal and a backup battery (eg. CR2032). Without a battery the device cannot possibly remember the time if it is powered off.

If your device is battery powered then you could conceivably just use millis(), however you would need to have a way of setting the time. The resonator used on a Uno is not as accurate as a crystal, however that might be good enough depending on your project.

If you are trying to save money I wouldn't use a Uno or Micro, but the bare chip, with a few extra components (eg. decoupling capacitors, a couple of resistors, and a crystal).