"There’s a certain level that is really hard to get through and most people who are just casually playing Donkey Kong, they can’t get through this one particular level."
- Will Forte, actor
"The average gamer on Donkey Kong will never get past the third elevator stage."
- Robert Mruczek, (former) Twin Galaxies Chief Referee
Level 4.4 (screen 13) of Donkey Kong is the portal, the enchanted wardrobe, the "first star to the left," the rabbit-hole, between the normal world and the world of the Donkey Kong adepts. It's a border, a gate... guarded by a merciless and terrifying onslaught of gigantic scissor-jacks.
Most will never get through this gate. Some will sneak past now and again by accident. But the player who has applied himself to the craft will be able to cross over at will at any time.
(Well, at almost any time. Even the elite masters die on "the springs" every now and then!)
To successfully climb the final ladder on screen 13 is to crack the surface and descend into the true depths of the game. It's a test. It probes all who approach in order to assess their worthiness, and clears out the everyday riff-raff.
It is my belief that this screen was specifically engineered by the game's creators to finish off any player who reached that point: to be so hard (though not too transparently impossible) that most would just give up, so that the machine could then be freed for another player (and another quarter).
It is well documented that this was the way arcade games were intentionally designed during the Golden Age—get them in, let them have a little fun, and get them out, in hopefully no more than 15 minutes.
It can't possibly be a coincidence that Level 5, which begins on screen 15—just one away from this infamous screen 13—is exactly the point at which Donkey Kong plateaus in difficulty. The developers stopped at Level 5 in terms of deliberate screen-specific gradations of gameplay, relying from that point forward on randomness and chaos to eventually kill the player.
Luckily, as difficult as this screen is for beginners to get past, it is at least (mostly) teachable, much moreso than some other aspects of Donkey Kong. The technique isn't easy to master initially, but it is little more than a technique, almost like a secret handshake with the machine (though with some optional variation).
Secret handshake or not, the tactic for getting through the springs is by no means a secret anymore, and information on how to do it is readily available to those who seek it.
At some point in the future I plan to publish my take on "the method", but until then, this YouTube video should help: