High school physics students, specifically.
Some might argue for none. I can see some reason in that. Many think of memorization of any kind as a burdensome task that kills all joy for miles around. Such absolute aversion may be an overreaction to bad instructional experiences from one's own past.
At the moment, I can see some value in having high school physics students internalize five constants:
Gravitational acceleration at the surface of the earth: g = 9.8 m/s^2
Elementary charge: e = 1.6 x 10^-19 C
Speed of light in free space: c = 3 x 10^8 m/s
Universal gravitation constant: G = 6.67 x 10^-11 kg m^2/kg^2
Coulomb constant: k = 9 x 10^9 N m^2/C^2
High school physics teachers all tend to think alike and readily agree on all maters pedagogical. So I'm guessing that if these are the values I think should be memorized, the rest of you agree by default. Am I right?