I believe the OP is describing a single slope ADC, which requires a very stable and accurate resistor and capacitor. I have a nixie tube DMM which uses this method. There is no voltage reference required (at least not in the ADC). Typically this methods limits accuracy to 3 digits or less, which is still an improvement on most of the analogue meters that it replaced.
In a multimeter, modifications to measure very low currents or high resistances is usually done in the signal conditioning stage, before the ADC. Some specialist instruments may integrate the input signal directly - e.g. some ion counters.