The oldest system that I know of were used by Old Kingdom Egyptians. Binary deben and other binary weight units were used in balancing statements that rounded off a missing 1/64 unit:
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (1/64)
To my knowledge corrections to the (1/64) error, and with 1/64 being the smallest unit did not formally change to 1/320 and smaller units until the Middle Kingdom.
In the Middle Kingdom prime number denominators began to be accepted such that zero round off was allowed in most cases.
Fibonacci may have have been equally accurate, though his Arab based method, that converted rational numbers to unit fraction series followed an algorithm, did not follow the finite unit sizes used by Egyptians and Greeks.
Read Sigler's 2001 translation of the "Liber Abaci" and other medieval texts for the specifics.