Do you mean physical constants as they are given in SI units or mathematical constants?

There's relatively few useful mathematical constants that aren't pure numbers, and the two most important ones (pi and e) are programmed into your calculator. The golden ratio is sometimes used, and the only other irrational number I see somewhat regularly are the zeros of the cylindrical bessel functions of the first kind--which is indicative of how rarely these are used in the first place.

As for physical constants? Really depends on what are you're studying more than anything. Physical constants are distinct in that they're unit-system dependent: speed of light is about 300 billion meters per second in SI, but 0.1 AU/minute. There's only two pure numerical physical constants I know of: Avogadro's Number and fine structure constant. If you're honestly using the fine structure constant, remembering it's about 1/137 is the least of your problems. Past that, I'd try to remain practical. If using SI units, you will remember the speed of light is about 3e8 meters per second, for example. I don't know the mass of electron or proton off the top of my head, however. You'll probably forget stupid constants like the Stefan-Boltzmann constant. But be aware what units you're likely to encounter, since that does change the numerical value.