If you're obsessed with base64 here is a fun thing for you:
A few years ago I randomly remarked that repeated base64 prefix converge to a fixed point. I tweeted [1] the first characters of that string out of amusement, without any additional details.
Then someone saw the tweet, found it funny, and did a write-up about it [2], and then there was an interesting discussion on Reddit [3]. There may have been one on HN too, I don't remember.