The following is taken verbatim from the MathWorld Wolfram page on Egyptian fractions:
An Egyptian fraction is a sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated to around 1650 BC contains a table of representations of $2/n$ as Egyptian fractions for odd $n$ between $5$ and $101$. … The unique fraction that the Egyptians did not represent using unit fractions was $2/3$ (The Penguin Dictionary of Curious and Interesting Numbers, Wells 1986, p. 29).
Well, I kind of find this surprising, since it is relatively easy to compute
$$\frac{2}{3}=\frac{1}{2}+\frac{1}{6},$$
which, according to the cited webpage [equation (4)], can be obtained using the greedy algorithm.
I checked the Wells reference, and it only has the following story to tell:
$2/3$ – The uniquely unrepresentative "Egyptian" fraction, since the Egyptians used only unit fractions, with this one exception. All other fractional quantities were expressed as sums of unit fractions.
I know that the answer to this question may be covered in some History of Mathematics book, but I currently do not have the expertise to gauge which authoritative sources to check.
MOTIVATION
It is currently unknown whether there is a number $m$ such that
$$I(m) = \frac{\sigma(m)}{m} = \frac{5}{3} = 1 + \frac{2}{3},$$
where $I$ is the abundancy index and $\sigma$ is the sum-of-divisors function. If such a number $m$ exists, then $5m$ is an odd perfect number, where $5 \nmid m$.
For more information on the exact connection between the Egyptian fraction decomposition of $1$ with odd denominators, and odd perfect numbers, I refer the interested reader to the following answer to a closely related question by MSE user Thomas Bloom.
Best Answer
They had a special symbol for 2/3, presumably because of frequent use, so there was no need to work out its representation. See chapter 7 of Annette Imhausen's Mathematics in Ancient Egypt: A Contextual History, Princeton University Press, 2016.