6(?) A&W teen burger patties, Plus a sliced hotdog
You can too, Just go to A&W and ask for an Animal Burger, It's not on the menu, but if you ask for it they will make it
  We used to eat these all the time when we were wandering around town at night all drunk
  
This is not my video, This A&W doesn't use a hotdog, These stupid kids can't even take a proper bite..... 
Edit: I just realized the one on the left is a girl......

 

Classic triple.  you have to eat an entire patty's worth of meat just before the burger can actually fit between the buns.

I've had a pounder from mcdonalds, which is 4 quarter pounders, for those who are stupid. 
so, 4.

UPDATE: 
Inspired by the multitudes of responses, I decided to make my trip to the local BK today a special one and ordered a Triple Stacker.  It was simultaneously a good and bad idea, as one can imagine such things are.  I don't plan on making a habit of it. 

3, multiple times...
Triple Whopper from BK once,
Triple Baconator from Wendys the second time...
 
mmm, Wendys is so good...

3.  Burger King used to make a burger that was just hamburger meat, bacon and cheese.  There were 3 patties on the basic one, but I think it went up to 5 for the super-meaty version.  That will clog your arteries right-quick.

1, Not a fan of eating stuff I can barely fit inside my mouth. Guess I'm old fashioned.

3. One of those stackers at Burger King. In all honesty, not as big as it probably should have been.

One.

One. If I want more meat, I'll just make a bigger burger. No need to make multiple small burgers to go in a single bun, when I can just make one large burger. And I've made some pretty large burgers in the past. 
 
Edit: actually, I tell a lie, I've had a double burger in burger king. But I don't really consider those burgers.

3. 
 
I've eaten the triple whopper and the triple whatever they are called at Wendy's.  I would not recommend it.  There is way too much meat and all you taste is the grease.  I definitely had an upset stomach after that.

2 ... and that was a BigMac.
 
Two much meat on a burger just doesn't taste well. What I do is just get another burger...

4...

i got six in one burger bar b queing at my house and it made me miserable.

The "four" in a Double Big Mac don't really count, right? If so, I'm a small fry at only 2.

The 4x4 from In N' Out (4 patties).  It's not that big though... I can definitely eat two of those, no problem.

3 at a wendy's, but it's not how many, it's how thick. I've had a burger that was 2 patties stacked, and it was more meat that 4 or 5 wendy's patties. FUCK that was filling. 

Four patties it was very delicious and I couldn't eat for the rest of the day.

3. I won't go so far as to say it was the last time, but it's deinitely like a "no more than once a year" kinda thing.

OVER 9000 !  
. . .c'mon guys, 4 pages and nobody said it? I'm kinda dissapointed  
Anyways the most I've had at a fast food place is 3, multiple times 
Homemade is 4

It's about quality not quantity. Oh, and two I think.

A quad from bk, but thats not that much for me, so I had another so does that count as eight?

3 square patties at a wendy's drivethru in my friends car. NEVER AGAIN

2.

2 in a Big Mac.. I feel ashamed

" 4  it was one of those BK stackers, they are kinda small but they stack em high. "

just had 6, made it myself though...

1 
 
I hate burgers, and fast-food in general. 
 
I have a feeling I am not welcome here.

This summer at a cook out I had four veggie burgers in between the same two buns. but the burgers were kinda skinny though.

I can handle two patties but 3+ I use a knife and fork. I know its blasphemy.

2. Haven't really had the chance to go for a 3x3 at In-N-Out.

Three on the Big Hardee when Hardee's in my area had it.

3, but it only came out to about 3/8ths pound of meat.

Octostacker, have it your way motherfucker

4x4 at In'n'Out. I was installing light fixtures at the time for a lighting company. I wrecked the toilet at one of the houses I am sure of it.

3 - tripple wopper at German Burger King, tasted not very good. It's just too many layers of meat (though the meat was fine).  

2
 
Much more is just excessive. I need to be able to fit it in my mouth for Christ's sake.

I think I've gone up to four, and a few times I've eaten two three-tiered beasts side by side. All this talk of burgers is making me starving, and it's 3am....

" This summer at a cook out I had four veggie burgers in between the same two buns. but the burgers were kinda skinny though. "