A compendium of quantities that happen to have nice numbers, for no reason

Fine structure constant

Defined as \alpha := e^2/ (4\pi \varepsilon_0 \hbar c ) .
\alpha \approx \frac{1}{137}.
Intriging, since it seemed for a time that it would be precisely the inverse of an integer number.

Actually,
\alpha \approx \frac{1}{137.035999084}

pH of water

The pH scale is logarithmic and inversely indicates the activity of hydrogen ions in the solution:
pH = −\log ⁡( a_{H^+} ) \approx − \log ⁡ ( [ H^+ ] ) ,
where [H+] is the equilibrium molar concentration (mol/L) of H+ in the solution. At 25 °C, it so turns out that [H+] ≈ 10^(-7) mol / L.

Viscosity of water

It so turns out that, at 20ºC, the dynamic shear viscosity coefficient is \mu \approx 10^{-3} Pa·s. Therefore, the kinematic viscosity coefficient is \nu \approx 10^{-6} m^2/s.

Far away place in general relativity

For a minimal distance R, a uniform acceleration a is given by a = c^2/R . This would be the normal acceleration needed to keep a particle traveling at the speed of light moving on a circle of radius R. If we want a close to g, the gravitational acceleration at the Earth, we would need to go very close to 1 light year. See Susskinds’ lecture on general relativity