The triple jump is the Snuggie of gaming. It's an admission that yes, we have alternative solutions to this problem, but we are choosing to be a slob at this point in time.
First and foremost. let's lay out just what the triple jump is (beyond a travesty). Generally found in platformers and platforming hybrids, the triple jump is accomplished by jumping for a third time while mid-air. Not once, not twice, but thrice. In other words, it's the ender of worlds.
Now, before we dive further into this travesty, let it be made clear that double jumps (that extra boost of air you get after your initial leap into the air) are no more excusable than the triple. The sole reason the triple jump deserves its own rant is that it is simply a fresh cat turd on an old and fly-ridden dog turd.
But how could I get upset over such a lack of realism in worlds that are just as surreal? Well, it's not the action itself that obtains my goat. As ridiculous as bouncing off an invisible bit of ground on a whim is, the triple jump does not stand out for its akwardness. Instead, it is a conceit made to counteract poor level design.
Sometimes, the "trip" is used as a means to block off areas or items from players. A sort of visual reminder that, yes, you must return to this area when you have unlocked the ability to look stupid mid-flight. Shitty thing is, we've already tried to get across using our less than viable double-jump, usually resulting in our death.
The real problem of the triple comes when it's used as crutch. The motto seems to be, "If your controls are slippery, dry off with a soft and absorbent triple-jump." Is there a worse feeling than falling back on your triple jump when you think you're going to come up short on that floating island, only to overshoot the thing and end up in the abyss? No. There never has been and never will be.
What makes me get bitch-faced is the question of why can't this be remedied by designing an area around a single, solitary jump? Is it a case of number-envy? Link does tons of junk without the ability to jump on command, why can't anyone else? Hell, even Metroid's alternative is a viable solution. What's the difference between a door that can only be opened with a certain weapon and the triple-jump? Not too much, really, just that one is a far more reasonable approach to the problem. You could even strap something to the protagonist's arm to zip him around and I woudn't bat an eye. Adding one more of the same thing just doesn't cut it. It's probably why Kazuma Kiryu of Yakuza fame doesn't end a flurry of punches with yet another punch. He mixes it up a bit, and our interest stays.
If nothing else, live by these words...
"If the game contains a triple jump, it's gonna be a triple dump."