Free electron said it best, but here's a simple application where the extra digits save you a lot of time, testing self discharge or quiescent drain on very power frugal devices.
On a standard NiMH battery, after you charge it full, connect it to 6.5 digit meter, and you can see in real time, the battery discharging in the uV range ~ 1uV/10 seconds or less.
Do the same with a LSD NiMH battery, and you'll see the same uV discharge in minutes to tens of minutes; once the battery has settled in to 85% of its rated mAH, it takes hours for a 6.5 digit DMM to detect a uV change. If you have a graphing DMM, you can actual see the rate of change, and graph the exponential decay.
The above is a very quick bench way to detect if a NiMH is actually LSD or not; with less digits, you'd have to wait longer to detect a change, taking up a lot of time.
Because sleeping devices consume uA or pA of current when OFF, its very difficult to get a sensitive ammeter to measure OFF current draw. But you can detect small drops of the battery supply at the uV level. For example, measure the battery drain to 6.5 digits in 1 min to detect self discharge rate, if at all, do the same with the low power device on OFF, with the same battery. If the drain is very small, increase the measurement time from 1 min to 10s of minutes or more.
The future of electronics is low power, devices, so in many ways the 6.5 digit is a meter of choice because of its wide availability and low cost; the next level is already 8.5 digits with 10x the cost, and there are not many 7.5 digit DMMs. At uV level, one has to use low level measurement techniques to avoid noise, stray fields, etc.,
Well if you are talking about a 6-1/2 digit meter, >95% of the time, you are talking about a bench meter. Bench meters, (well atleast the ones worth considering), are intrinsically stable. Just take a look at all of their fancy precision references and Caddock resistors you see. Yes, most of them are darn stable.
The question is more on the lines of... Ex. 2.55554. For practical matters, I'm curious to know when you'd need to know the fate of that last digit. I'm not really trying to make a point. I honestly want to know what application you need such precision for
And if we really just want a 6 1/2 digit meter so we can have a better 4 1/2 digit meter...why can we not make a better 4 1/2 meter?