Below are all the numbers expressible in Korean(a lot).
You can see that the highest number is 10^68, which consists of 68 zeros.
Most of there words have disappeared because of lack of usage.
However, some still exist in everyday talk, only that they don't represent actual numbers anymore(ex-'찰나의 순간' stands for a very short period of time, and you can see that '찰나' originally stood for 10^-18, a very small number.
Also, like many other Korean words, there are actually two kinds of expressions for numbers in Korean: the'순우리말'system and the'한자어'system.
The answer above explained about the former system, and this answer consists of the latter.
청정(淸淨) 10^-21
허공(虛空) 10^-20
육덕(六德) 10^-19
찰나(刹那) 10^-18
탄지(彈指) 10^-17
순식(瞬息) 10^-16
수유(須臾) 10^-15
준순(逡巡) 10^-14
모호(模湖) 10^-13
막(漠) 10^-12
묘(渺) 10^-11
애(埃) 10^-10
진(塵) 10^-9
사(沙) 10^-8
섬(纖) 10^-7
미(微) 10^-6
홀(忽) 10^-5
사(絲) 10^-4 = 0.0001
모(毛) 10^-3 = 0.001
리(厘) 10^-2 = 0.01
분(分) 10^-1 = 0.1
일(一) 10^0 = 1
십(十) 10^1 = 10
백(百) 10^2 = 100
천(千) 10^3 = 1000
만(萬) 10^4 = 10,000
억(億) 10^8 = 100,000,000
조(兆) 10^12
경(京) 10^16
해(垓) 10^20
시(枾) 10^24
양(穰) 10^28
구(溝) 10^32
간(澗) 10^36
정(正) 10^40
재(載) 10^44
극(極) 10^48
항하사(恒河沙) 10^52
아승지(阿僧祗) 10^56
나유타(那由他) 10^60
불가사의(不可思議) 10^64
무량대수(無量大數)10^68