A former department chair at BYU, Wayne Barrett, would always amaze grad students by his vast knowledge of mathematical constants, like the radical form of $\cos(2\pi/5)$. I've never memorized anything beyond some digits of $e$, $\pi$, and $\sqrt2{}$, but I know many famous mathematicians could pull out constants and perform calculations by hand extremely well.
For those of you who began working before computers became big, what kind of things did you memorize?
I would suggest calculators are more important to this than computers. It depends a lot on what you want to calculate. I didn't sit down to memorize constants, but like to do mental calculation, much of it involving naturals or estimates, so the following is my list. In the spirit of the question I will use $=$ where $\approx$ is more accurate:
$(1+x)^n = 1+nx$ for $nx \ll1$ Probably the most important, because you can correct other things with it.
All perfect powers up to $1000$
Powers of $2$ up to $2^{16}=65536$
Primes to $100$
$\log_{10} 2 = 0.30103$ or the similar $2^{10} = 10^3$
$\ln 10 = 2.3, \ln 2 = 0.69$
$1$ radian $=57.3$ or $60$ degrees (this so you can use the trig functions of the next line)
trig functions for $30,45,60$ degrees (not $15,75$, they don't come up for me)
$\sin x=x, \cos x=1-\frac {x^2}2, \tan x = x$
square roots of $2,3,10$ to three places
decimals for $\frac 1n$ for $n \in [2,12]$
triangle and Fibonacci numbers up to $100$ or so
Finally, $\pi = \sqrt {10}= 3$
Somebody recently was surprised that we don't all know ${49 \choose 6}$ because of lottery problems, but I don't.
I think this has little to do with mathematical ability.
