Does the size of bubbles in water brought to a boil ever vary as a function of the amount of water?

Bubbles are formed when the pressure in the fluid is less than the saturated vapor pressure of the liquid at that temperature, modified by surface tension effects.

Surface tension actually increases the pressure inside a gas bubble in water; the approximate increase in pressure is $\Delta P = \frac{2\sigma}{r}$ (see for example this earlier answer). The pressure can be lowered by adding impurities in the liquid that can act as nucleation sites.

For a bubble to start growing, the vapor pressure has to overcome the "net" pressure barrier. When you have a larger volume of liquid, and you are heating it from below (like a pan on a gas flame), then the pressure will be higher if the liquid level in the pan is deeper. The pressure increase at a depth $d$ is $\Delta P = \rho g d$.

This means that the liquid at the bottom has to be hotter in order to start boiling; it also means that as the bubble rises in the liquid, it will more easily grow (both because the vapor inside can expand as the pressure is less, and because, if the liquid temperature is uniform, the lower pressure inside the bubble will allow for additional evaporation into the bubble.)

Finally, as I mentioned in my comment, in the limit of very small volumes of liquid it's possible that the production of steam at the surface of the liquid drop is such that a bubble is never formed: instead the vapor escapes along the bottom of the drop, which ends up suspended on a cushion of vapor.