When calculating the RC constant for an extremely simple circuit, say a battery connected directly to the poles of a capacitor, is R essentially zero with a correspondingly fast t? Or does the internal resistance of the battery and/or the capacitor play any role in R's value?
(I'm actually not going to do this precisely, so no need to warn me ;-) )

 

When calculating the RC constant for an extremely simple circuit, say a battery connected directly to the poles of a capacitor, is R essentially zero with a correspondingly fast t?

Only for a perfect capacitor connected to a perfect battery, with no internal resistance in either.

Or does the internal resistance of the battery and/or the capacitor play any role in R's value?

Yes, the battery and capacitor internal resistances in series with any resistance in the connecting wires, comprise the R.