This is my absolute favorite physics experiment; I actually wrote a miniature essay on my lab report containing, in essence, the following:
Millikan's experiment is amazingly indirect. You don't directly measure the charge of the electron, nor do you compute it from an equation (like, say, the one relating electrostatic force to charge, though you do use this equation to deduce the non-unit charge on each oil drop). You don't even deduce it by fitting a curve to a set of data points. You do deduce it from a frequency graph, but it is, again, not any kind of fitting of the actual frequency data; rather, you identify spikes in the frequency counts and mark off the corresponding charges. Then you find their greatest common divisor and argue that your data was random enough that this must be the charge of one electron (since it is vanishingly unlikely after enough droplets that each one contained only, say, even numbers of electrons). It is actually a simple form of image recognition.
You don't need to luck into one-electron droplets. You just need to keep measuring until you have an unambiguous set of frequency spikes at sufficiently many different charges and apply (what?!) a number-theoretic computation.
The business of taking the greatest common divisor is a bit tricky in the presence of measurement errors: after all, the charges you find are not only subject to error from your lab equipment but also from your identification of the center of each spike. You can assume the charges are all integers by converting your limited-precision floating point numbers to fixed point, but those integers are almost certain to have a GCD of 1. For example, if you measure charges of 201 and 302, you'll find that the fundamental charge is not 100 (which is the obviously correct answer) but rather 1.
You can, of course, eyeball it: say, you can take various ratios and fit them to nearby rational numbers with a small common denominator (in the above example, the ratio is approximately 1.5025, so you easily find 1.5 = 3/2 as a likely "correct" ratio). A better way is to use an "error-tolerant" version of Euclid's algorthm. In short, proceed as usual by dividing (with remainder) the smallest number into all the others and repeating, except that rather than waiting for all the remainders to be 0 (indicating that the last remainder was the GCD), you wait for them all to be "small" in some sense. Say, an order of magnitude smaller than the previous one.
Take the above example: Euclid's algorithm gives you the following sequence of remainders: 302, 201, 101, 100, 1 (each one is the remainder of the division of the previous two). This suggests that 100 is the correct GCD, as indeed it is. Amazingly, the algorithm actually wiped out the measurement errors and got the exact correct GCD; I don't know if this kind of "focusing" effect is typical or if I just happened to use the right numbers.
This only increases my love for this experiment.