Was the discovery of six exoplanets around one star as "easy" as counting six peaks in the FT? The phys.org article Scientists make huge dataset of nearby stars available to public describes the release of a publicly accessible database of Echelle radial velocity measurements; The LCES HIRES/KECK Precision Radial Velocity Exoplanet Survey. See also Keck's HIRES home page.


  For two decades, these scientists have pointed HIRES at more than 1,600 "neighborhood" stars, all within a relatively close 100 parsecs, or 325 light years, from Earth. The instrument has recorded almost 61,000 observations, each lasting anywhere from 30 seconds to 20 minutes, depending on how precise the measurements needed to be. With all these data compiled, any given star in the dataset can have several days', years', ore even more than a decade's worth of observations.


This part caught my interest especially:


  "We recently discovered a six-planet system orbiting a star, which is a big number," Burt says. "We don't often detect systems with more than three to four planets, but we could successfully map out all six in this system because we had over 18 years of data on the host star." (emphasis added)


For very simple cases of one, or maybe two planets with minimal interplanetary gravitational interaction, a Fourier transform of a nice, long, continuous radial velocity measurement would show two main peaks, and possibly other artifacts. If the stellar motion induced by each planet were similar magnitude, the analysis might be fairly simple.

But for the six planet case mentioned in the quote (I don't know which one it is), and patchy time coverage (it's a survey) how was this analysis done? Peaks alone? Or just throw it to a supercomputer simulation of every possible combination and let simulated annealing run for a month?

Or was there some 'detective work' involved as well - assumptions, limitations of the fitting space, or even inclusion of other data from outside the study?
I suspect that the record holder (as of 14/2/2017) is HD 10180 which has at least 7 planets and possible evidence for as many as 9.

Lovis et al. (2011) announced the initial discovery based on 190 radial velocity measurements taken over 6 years. The precision of the measurements was 0.3-0.9 m/s.

Section 4 of that paper describes how they go about finding the planets in the data. It is a hybrid of fourier and fitting methods. As each peak in the periodogram is found it is added to a model which is then iterated towards a best solution. 

Successive planets reduce the rms scatter around the predicted radial velocity curve. Ultimately a judgement has to be made whether the improvement in the fitting statistic justifies the addition of another planet (and more free parameters) to the model. The final rms after adding 7 planets was just over 1 m/s, which is worse than the expected precision, but which Lovis et al. attribute to (probable) radial velocity jitter due to stellar activity. The orbital model is then refined to include the effects of planet-planet interactions and tidal forces.

The radial velocity amplitudes attributable to each planet range from 0.8-4.5 m/s. The most marginal detection has the smallest amplitude, but the shortest period (more cycles and hence easier to detect smaller amplitudes). 

A later paper by Tuomi (2012) used a more conventional Monte Carlo Markov Chain in a Bayesian framework to fit a model of $n$ non-interacting planets to the radial velocity data. Again, there is lots of discussion (see section 3.3 and 3.4) of exactly how many planets are required to fit the data. Tuomi claims there is strong evidence for an 8th and 9th planet in their analysis.

There are a number of important assumptions made in these types of analysis. The main one is that you have to assume some sort of model for the background noise and it is often assumed to be Gaussian and non-periodic.