I like fermenting on-top of the previous batch’s yeast cake. But on the other hand, with all the care we put into cleaning and sanitizing, I’m worried about contaminating the new batch. I rack them to kegs on brew day, but my primaries will have been going for over a month when that day arrives. One of them will be a thirdy, and the insides are plastered to the rim with yeast residue. No worries, or am I doing something wrong?
If the beer you rack isn’t contaminated, there’s no reason to believe your slurry is.
Contamination shouldn’t be a problem (assuming your previous batch was sanitary) but over pitching can be a concern, and certainly the build up of dead cells over multiple repitching can be. You will get more consistent results by pitching only a part of the slurry. Check the pitching calc at www.mrmalty.com to get an idea how much slurry you should be pitching.
Except that an unnoticeable, below threshold, small contamination in one batch can turn into a major contamination in the next batch. If you keep doing it I’d expect to lose the occasional batch unless you’re very lucky/good. 
Plus if you keep racking on to the cake from batch to batch you’ll eventually get off flavors from autolysis and you’re using a lot more yeast than you should. At some point you need to split the yeast cake and wash out some of the trub. And you might not want to rack a bitter on to the cake from a stout, it will add a lot of color.
It can be done and done well, you just have to be aware of potential issues so you can avoid them. 
If you follow good sanitation practices there should be no cause for alarm.
That being said, I would recommend using a clean and sanitized fermenter (if available) and as majorvices has indicated, measure a calculated quantity of slurry and pitch it into the fresh wort. If you don’t have another available fermenter then you should be okay to use the current one but estimate your required yeast and remove any yeast as required.
I would not use the same fermenter (without cleaning/sanitizing) for more than two batches for reasons that Tom has stated, the reason being that your potential for infection will increase with from batch to batch and it’s probably not worth the risk IMO.
All of the above is spot on and Tom said it very well. Instead of over-pitching I prefer to divide up the first yeast-cake and store that yeast. It makes me nervous to pitch on top of yeast that has been through more than several batches. This way I can pitch with a little more confidence that successive generations of bacteria haven’t gained a toe-hold in my slurry; and that the yeast haven’t been beat up so bad they won’t propagate and perform properly.
Any good reference on how to divide and wash the slurry?
I’ve never washed yeast. However, a funnel and plastic (PET) soft-drink bottles work well. Sanitized of course. I actually prefer the little 8oz water bottles- the kind that fit in school lunches. These get stored in the fridge. If they sit for a while I’ll pitch more than one, but a fresh bottle of yeast should be plenty for an average batch of beer.
http://www.homebrewtalk.com/wiki/index.php/Washing_yeast
I’m not comfortable with using glass. But hope this helps.
I was just listening to this Brew Strong episode this morning because I’m going to wash my yeast for a new beer in about 2 weeks.
There’s a distinct difference between washing and rinsing the slurry. The latter involves diluting, swirling, settling and decanting. Always a good idea. Washing involves acid…I don’t plan on going there anytime soon.
Your avatar suggests you may have- at least once… :
I don’t wash or rinse my slurries. I simply divide by eye by pouring intpo 2 sanitized plastic containers with snap on lids.
+1 - I saw no appreciable benefit when I rinsed my slurries.
in full disclosure however, I do have the ability to dump trub, dead yeast and other waste during high krausen which probably means less junk in my slurries.
ymmv.
I don’t wash or rinse my slurries either. I simply skim off the top half of the yeast cake as to avoid the trub layer and store in a sanitized container until further use. I like to store my yeast at 34F.
Tubercle brews a new batch, racks existing finished beer into the keg, scoops out ~2/3 of the trub and then fills the fementer. No problems so far.
Plus there is the Frankenyeast that has been going for about a decade now. It lives in a quart jar in the fridge. It is let out every once in a while to play.
I have conicals as well, and do pull from the middle of the slurry. But when I was using carboys full time I found the same as Denny, washing didn’t make an appreciable difference. That said, I was never using more than 3 or 4 generations. If you go higher than this I think washing becomes more important for removing dead cells and trub.
Did you name your yeast?
If not could I suggest Herman?
They’re just called my little buddies (some will get the pun, some won’t) but it is time for a formal name - Herman it is.
A friend of mine has a bread yeast culture going in the fridge…it is over 10 years old. It is known as “Bob”.