I've previously mentioned Sanjoy Mahajan's Street Fighting Mathematics. (Yes, that's right, almost the entire sentence is links, deal with it.)
One thing I didn't mention is approximating logarithms using musical intervals, from that course. We all know 210 and 103 are roughly equal; this is the approximation that leads people to use the metric prefixes kilo-, mega-, giga-, tera- for 210, 220, 230, and 240 in computing contexts. Take 120th roots; you get 21/12 ≈ 101/40.
Now, 21/12 is the ratio corresponding to a semitone in twelve-tone equal temperament. So, for example, we know that 27/12 is approximately 3/2, because seven semitones make a perfect fifth. So log10 3/2 ≈ 7/40 = 0.175; the correct value is 0.17609... Some more complicated examples are in Mahajan's handout.
You might think "yeah, but when do I ever need to know the logarithm of something?" And that may be true; they're no longer particularly useful as an aid for calculation, except when you don't have a computer around. But I often find myself doing approximate calculations while walking, and I can't pull out a calculator or a computer! (To be honest I don't use this trick, but that's only because I have an arsenal of others.)
Is this pointless? For the most part, yes. But amusingly so.
The method is supposedly due to I. J. Good, who is annoyingly difficult to Google.
Oh, and a few facts I find myself using quite often -- (2π)1/2 ≈ 2.5, e3 ≈ 20.