It’s not too hard to come up with these:
1/271 = 0.003690036900369003690036900369
1/369 = 0.002710027100271002710027100271
1/4649 = 0.00021510002151000215100021510002151
1/2151 = 0.00046490004649000464900046490004649
All you need is a pair of numbers that multiply to 999…999, for some number of 9s.
Ah good, I was worried it would be something hard, like factoring large numbers… 
UIAM, the repeating period (in any base) of a rational number p/q will be a divisor of Euler’s totient function φ(q). For example φ(49) = 42 and, in base 10, the repeating string attains its maximal length of 42:
1/49 = .020408163265306122448979591836734693877551
020408163265306122448979591836734693877551
020408163265306122448979591836734693877551
020408163265306122448979591836734693877551 …
There is a vague roundabout connection! From the above-linked article:
But not all are “pretty”. Take another pair of factors of 999: 9 and 111.
1/111 = 0.009009009 …
1/9 = 0.111111111 … which you have to “see” as .111 111 111 … .
That was a really good video. I’m going to watch it again later in detail in order to fully absorb the moleeds theory into my thinking.
I find the area of mathematical ratios and rational numbers fascinating, especially since through sound/music we are able to hear them.
Many years ago, Martin Gardner had a great Mathematical Games column about the related topic of “cyclic numbers”, for example the integer 142857, which is the repeating part of 1/7 (see the post by Chronos above). If you multiply it by any integer less than 7, you get a cyclic permutation of the digits.
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142
You can find more of these by looking at the decimal expansion of reciprocals of prime numbers. The example of 1/17 was given above by Dr. Strangelove.