Alright, kids. Gather 'round.
Grandpa Flexington is back on his rocker. - And to save you some time, this is basically the long winded version of most of what was already said here, and way more information than you're asking for.
This is a bit of a simplification of the process, but it is essentially how it works - Proc rate is normalized over time, functionally making procs per minute a valid measurement whether it's the devs intended goal or not. Procs function in much the way you think they would logically, so a lot of this is going to seem self explanatory. On each given combat round they have a percent chance to fire. However, the game adjusts this expected proc rate to average out, over time, to X procs per minute relative to your current state (that is to say, if your attack speed is reduced, the proc chance per swing increases to still average the same over time, and the opposite is true for haste/hundred hands effects). After all calculations are done, it applies the appropriate percent chance to fire to each individual swing (double and triple attack are single swings as far as the game is concerned) - So to simplify that. Procs are intended to trigger X times per minute regardless of attack speed, and that is adjusted in real time. No real surprises there.
That was why
earthshaker shakerpaging worked back in the day. You would slow your own attack speed to the point that it needed to proc every few swings to reach its intended proc rate. When you gathered up 100+ mobs and hit rampage aa, the proc would register on nearly every single enemy in range and, due to the aoe nature of earthshaker's proc, would just alt+f4 the entire lot of them. Rampage can no longer cause a proc on a mob that is not your current target, thus killing shakerpaging.
So that being said, if you have an expected one proc per minute and have two combat rounds per minute, every other swing should proc, mathematically. However, there's nothing stopping you from proccing two combat rounds in a row and then not proccing twice in a row, because it's still a flat 50% chance per swing in that scenario. But eventually, over the life of that proc, it will average one proc per minute, ignoring new modifiers being added to the equation. Just like a slot machine with a 50% payout, you can get lucky and win more than the expected rate in the short term... But it's going to come back and you'll have 'bad luck' later. If you parse for two minutes, you might see a 75% proc rate. Amazing!... But if you parse for two days, you'll see that 50% holds true. Once again, probably no real surprises here.
As to what Many_Whelps said about things that are rate limited, I don't know of any weapons or spells with a rate limited proc other than banestrike (that's not to say they don't exist, I'm more out of the loop than I've ever been, but I'm not aware of them). This rate limit can be seen in
Raidloot's spell data as 'Proc Timer: '
As aspire2008 mentioned, once upon a time you could only fire one proc from each category (weapon, aa, buff) per combat round. This changed in
July, 2015, and just like they said, all procs are now totally independent of each other. There is nothing stopping every proc from firing off in a single combat round. And, in fact, you can have some fun with this by stacking as many procs as you can on a ranger and hitting Imbued Ferocity so they all fire every combat round. Makeshift fireworks! (Don't lose a finger)
The base proc rate on most weapons is 2 per minute. Proc rate is modified by the weapons innate proc rate, dex, hdex, weapon affinity aa, combat effect mod 2 (which now comes from hdex), and there are a few various spells which modify proc rate (like monk and shaman spires). 10 points of combat effect is roughly equivalent to 1 rank of weapon affinity aa which is a 10% increase to the base proc rate. Sancus has some
good information on proc rates here. He is a mage so it's missing information about weapon affinity, but you can just add that in to his formula as 10 points of combat effect per rank and it still basically works. Unfortunately, his formula does not account for hdex which we know is part of the equation because of a direct quote from a dev.
"Proc chance is a function of your base hardcapped Dexterity / 35 + Heroic Dexterity / 25 " - Elidroth, Feb. 13, 2014 Dev Chat
In the end basically all this mean is proc rates are independent from attack rate because they're normalized over time. The important take away for how it functions to us (at least relating to your question) is that on any given swing, assuming no variables have changed, the percent chance to proc stays the same. It's totally possible to get 10 procs in a row at a 10% proc chance with god tier luck (and an infinite amount of monkeys).
TLDR: Ok, now with all that out of the way I can actually answer your initial question. Save for the possibility of special exceptions existing, there is functionally no limit to the amount of procs you can get from a weapon, up to firing every single swing (as was demonstrated nearly 20 years ago with shakerpaging... see, I told that story for a reason), but due to the limited amount of increased proc modifiers, you're probably only ever going to hit 100% proc rate while actually using a 100% rate ability like ranger's Imbued Ferocity. But stacking hdex is as close as you're going to get otherwise.
I won't touch on the debate about what stat a raid knight should focus on because that's another raid boss encounter with 'ole Flexington and it's still debatable.
However, Pets are a different story...
I believe pets can now also fire procs independently of each other as well. I am not in a position to test it currently to be 100% sure, however, so don't take that as gospel.
But yes, this was a totally legitimate point, particularly in regards to pet tanking, in the past. The epic, with no rune proc, would take priority over iceflame and occasionally cause you to take two additional full damage hits you otherwise wouldn't have.