Fractions that have interesting, fun or noteworthy decimal expansions I'm looking to discover more fractions that have interesting* decimal expansions. (I'm asking out of curiosity, there is no particular academic reason as far as I'm concerned).
Here are a few examples:
$\dfrac{1}{99}=0.010101\dots$
$\dfrac{1}{999}=0.001001001\dots$
(and so on...)
This post talks about:
$\dfrac{1}{243}=0.\overline{004115226337448559670781893}$
This Numberphile video talks about how:
$\frac{1}{999^2}=0.00000100200300400500600...$
'generates' the 3-digit integers (except $998$). Similar patterns arise with $1/99$, $1/9999$, and so on.
*I realize that 'interesting, fun or noteworthy' might make this question a bit open-ended or subjective, hence the 'soft-question' tag. Then again, I find it difficult to be more specific about what I'm looking for.
 A: Given any repeating decimal expansion, you can find the rational number that generates it. For example, let's say I think that $.\overline{02040608}$ is interesting, then there is a process of calculating the corresponding fraction:
$$x = .\overline{02040608}$$
$$10^8x = 2040608.\overline{02040608}$$
$$10^8x - 2040608 = x$$
$$x = \frac{2040608}{10^8-1}.$$
You can generate any periodic decimal expansion you want via this method.
Edit: This method also illuminates why we see stuff like 99, 999, 9999 so much in these `interesting' decimal expansions.
A: The number $$1/89 = 0.0112359550561798\ldots$$ is interesting as the digits form the Fibonacci sequence $1,1,2,3,5, 8, 13, 21,$... at least for a while.  If you want more terms of the sequence you can use, e.g., 
$$1/9899 = 0.000101020305081321\ldots$$
which gives you the two-digit Fibonacci numbers or 
$$1 / 998999 = 0.000001001002003005\ldots$$
which gives you the three-digit Fibonacci numbers.
Why does this happen? I cannot tell you; it is a secret you must discover yourself...
