I think it comes down to creativity vs. time management.  My rule of thumb, I only work with 3 takes or 4 at the most.  If I'm having to decide between comping sections on10 different takes, my head would explode.
 
When I lay down drum tracks, I know I am most likely to screw up coming in or out of a fill.  This is where spontaneity, can kill you.  Getting too energetic can cause me to rush through a fill.  I like what Vastman said, lay down something simple then use comping if you want to experiment.
 
As to the original question, yes I use most of the first take, but then it's mostly because I know where I've screwed up on the first take, and what sections went well, and what I need to concentrate on in the proceeding takes.
 

My record is 39 takes of a guitar track, though I seem to always get the keepers around takes 4-7.
Oddly enough, as a photographer my keepers often seem to be frames 4-10.
I put this down to fidgeting and beating my mind into obedience for the first few minutes...then having boredom, backache or a blood-sugar dip after about 45 mins.

Instruments I tend to stop and re-do as I'm going along if I mess up, maybe edit midi a little at the end to tighten a few bits up, but generally by the time I've gotten to the end of the song, that's the finshed track. Main vocals I usually do 3 or 4 takes all the way through, depending on how well I think I did, but usually end up using most of take 1 or 2 with the odd phrase from 3 or 4. BGV I am usually making up as I go along, so that will be more of a stop/start thing. I don't usually do any correction on BGV's and will just record until it sounds reasonable. Haven't ever thought about why I do that, I just do. Never really thought about there being a pattern to the takes before, but it's true in my case and interesting to see that others find the same thing.
 
Haven't really got to grips with the new comping method yet, still doing it the old 8.5 way (Sorry Craig!).

The great thing about the comping feature is that every take now I just keep playing all the way through a song whether I screw it up or make a minor mistake.
 
I used to stop after every mistake and start over again.  ( I make a lot of mistakes...)
 
Usually I use most of take 1 or 2 and then fix the mistakes with take 3 or 4.
 
Now that I know how easy it is to comp, I am much more relaxed on every take....thanks cake !

Featherlight
There seems to be three types of session performers for me.
1: come in and nail it in the first couple of takes and there's lots to work with
2: hunt around for the part and then slowly craft it into a working line, later takes are the most refined
3: kind of like Nuke LaLuche in Bull Durham, all over the map and I end up comping the crap out of it and 'building' the part lol!

 
These 3 types describe it really well!

I see mostly type #1 and #2 (thank god) ...
 
I reckon, #1 is also related to who wrote the material i.e. if you are a skilled musician and you wrote the part, you're deeply into it from take #1 and the spontanous first few takes are the magic ones ... sometimes you end up tracking more to make the artist happy, and then comp out of the first 3 takes in your lonesome editing session ...
 
If you write for other people and they come in to record your material, you have a type #2 situation: it might take quite a while to put them into the right picture, mood, feel, intensity, ... so you end up recording a 'zillion' takes, each one being better ... if that happens you need your motivation skills to squeeze everything out of the performer and stop as soon as you reach the plateau ... then I use the last 3 takes (the last being one being the convincing performance that made you stop recording, the others being backup for things you discover when cleaing the tracks)
 
If you have type #3 in the rec room, you might as well stop and hit the bar early - I don't believe that a fully comped performance will go thru as a real performance; even if you technically might not hear it, the audience will feel that something is not quite natural
 
So bottomline for me: there's no point comping out of more than 3 takes really (the exception may be a multi-mic'd drum session with very sensitive timing)