My textbook says:
Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass.
Does this mean we can have arbitrarily small masses? In particular, can we have masses smaller than the electron?
The answer to "does it follow from non-quantization of mass that arbitrarily small masses are possible?" is no. If the lightest particle has rest mass $m>0$, then there is no system with a mass in $(0,m)$, but the mass spectrum would be continuous above a certain point, so mass wouldn't be quantized. For example, assuming there's no strong force like electromagnetism between the particles, the rest mass of a system of two of the particles would be roughly $2m\sqrt{(1+γ)/2}$ where $γ$ is the gamma factor of their relative speed, so any mass in $[2m,\infty)$ is possible. (That's a bit of a cheat, because the particles will fly apart unless you confine them in a box, which would also have mass. But by the strict definition of rest mass it's true.)
In reality, the lightest particle is not an electron or neutrino but a photon. A system of two or more photons can have any nonzero rest mass (with the same caveat as above).
Your textbook's statement is correct. If you consider the case hypothetically, you can have infinitesimally small masses, but in practise, the smallest non-zero mass of a particle that has been measured is the electron mass, 511 keV. There is no limit to the size of a mass that can be measured.
Note: One of the comments say that neutrinos seem to have masses smaller than that of electron which is absolutely correct but we have not measured it to a satisfactory extent.
