I might be different, but 355/113 is very easy to remember and is not 7 unique segments of information. I just think "double the odds"
113355
We know we want a fraction, not a single number, so split down the middle:
113/355
And we know pi won't be less than 1, so flip it: 355/113. Knowing that the digits sequences are doubled lets you do some cheap, mental run length encoding. In a case where you need a hand calculation, spending 3 seconds to re-derive the sequence seems tolerable.
I came back here 6 hours after reading it to confirm that it is still stuck in my brain, I am guessing permanently. Although I already knew Pi to 6 decimal places so its not like I am gaining a lot of accuracy out of this.
113355
We know we want a fraction, not a single number, so split down the middle:
113/355
And we know pi won't be less than 1, so flip it: 355/113. Knowing that the digits sequences are doubled lets you do some cheap, mental run length encoding. In a case where you need a hand calculation, spending 3 seconds to re-derive the sequence seems tolerable.