I have read many post fermentation opinions/practices. Some move it to secondary after a week, some wait 3 days after fermentation is done and then bottle condition; and many leave it sit on the yeast cake for a period of time to let it "clean up after itself". I have heard at least one person even report that they've left beer on the cake for a few months! To me this all sounds like something I'd need to experiment with myself but is there any science or common knowledge suggesting a time after which the yeast would be very likely to be finished "cleaning up"? I want to know so that I could determine a maximum length of time for any experimenting - ie I don't want to leave beer on the cake for 3 months if that would likely be a waste of time in most people's experience/opinion.
According to info I got from John Palmer for an upcoming article I wrote for BYO magazine, the cleanup happens concurrently with fermentation. Here's the quote..."Yeast have 3 phases in their life cycle: Adaptation, High Growth, and Stationary. (See Yeast by CW and Jamil) They do not have a maturation phase where they clean up byproducts. Adaptation phase is where they take in oxygen and build sterols and other lipids, assess the sugar composition and build enzymes, etc. Once those activities are done, they start the High Growth Phase, eating and reproducing. The number of cell divisions is limited by their lipid reserves they made during Adaptation. These reserves are shared with each daughter cell. When those lipid reserves are exhausted, the cell stops reproducing. In addition, when those reserves are exhausted, the cell is old and cannot eat or excrete waste efficiently across it’s cell membrane. A yeast cell typically can reproduce about 4 times during a typical fermentation, after that it is old and tired and tends to enter Stationary phase where it shuts down most of its metabolism and flocculates, waiting for the next batch of aerated wort. Stationary phase is essentially an inactivity phase, resting on the bottom.
Like I said, no conditioning phase as far as the yeast are concerned. Byproducts can be consumed at any point during the high growth phase, but they are a lower energy source than sugar, so guess what? Byproducts are not a biological priority. The brewer therefore needs to plan his pitching rate and fermentation conditions such that the yeast run out of fermentable wort sugar before their lipid reserves are exhausted and they go into stationary phase. Now you have a majority of vigorous yeast that have only undergone 2 reproductions (for example), the sugar is gone, and they are still hungry, so they turn to acetaldehyde and diacetyl as alternate energy sources and maturate the beer. You can help this by doing a diacetyl rest by raising the temperature a few degrees after the first half of fermentation, to keep the yeast active and eating. Where in the fermentation? after the first half, 2/3 to 3/4, when most of the attenuation has occured and raising the temperature is not going to cause rampant growth and the off-flavors associated with it. "
There are a couple things yeast will clean up after fermentation.
We do sensory analysis for two off-flavors during fermentation: diacetyl and acetaldehyde.
Diacetyl occurs when alpha acetolactate (AAL), a byproduct of fermentation, is exposed to warm temperatures (however conversion happens slowly at cool temperatures). Diacetyl is classic fake-butter smell that many people are sensitive to. The yeast will consume diacetyl towards the end of fermentation.
When our beers have reached a stable terminal gravity we do what is commonly called a forced-diacetyl test. Two samples are taken from the fermenter. One is heated to 140° F to force the conversion of AAL. Both samples are then chilled to drinking temperature and compared. I am blind to diacetyl, but some of my sensory crew are sensitive. If the beer fails diacetyl we let it sit another day or two.
Acetaldehyde is produced during fermentation as an intermediate compound in the conversion of glucose to ethanol. During a normal fermentation the yeast will consume the compound. Its signature green-apple flavor is a sure sign of an anemic fermentation.
Acetaldehyde will clear out given time with healthy yeast.
Leaving on the yeast cake for months, is far from ideal and risks autolytic flavours making their way into the finished Product. You can condition in a secondary for a number of months, but you want to do this off the main yeast cake.
Regarding cleaning up by products 3-7 days after FG is reached is more than enough to clean up the main by products left from the main fermentation.
EDIT - Apologies if I made this 'clean-up' sound like a different phase in the yeast life cycle that was not my intent. I simply intended to highlight that even after Final Gravity has been reached the yeast are still metabolically active and will break down metabolites, in the absence of sugars, into non-flavour active compounds over the course of a few days.
EDIT - Even when yeast enters G0 - stationary phase it is still metabolically active  even if it has flocullated out, it will continue to affect the flavour of the beer, until it dies. Even then it will further affect the flavour of the beer by releasing the cell contents into the solution as the cell walls rupture.
If you are then bottle conditioning the product will continue to change and develop until the day it is drunk (EDIT: for myriad reasons; some yeast related). 
 J. Inst. Brew., May-June, 1986, Vol. 92, pp. 213-219 213 CENTENARY REVIEW THE BIOCHEMISTRY OF MATURATION - Charles A. Masschelein