Update - May 11th, 2016: This post was written in 2011, when neither the player base, nor point-pressing theory, had reached full maturity. Only two players had cracked 1.1 million points at the time, and none were anywhere close to 1.2. Almost five years later, Wes Copeland (a player we hadn't even met at the time this was written) effectively brought the world record chase to the "end" referred to here with a score of 1,218,000. The post will remain, for historical purposes.
Since Donkey Kong terminates itself at the beginning of Level 22, and since a limited number of points are available on each screen, the game necessarily has a score ceiling at which no more points can possibly be squeezed out of its 117 screens.
The ceiling has been underestimated in the past, most notably by Tim Sczerby, who once voiced doubts (which have since been soundly disproven) that even a million points could be achieved on unhacked ROMs. Current prevailing wisdom among the game's top players asserts that the practical ceiling is approximately 1.2 million points. In 2008, former Twin Galaxies referee Robert Mruczek prepared a detailed mathematical analysis proposing 1.3 million as the maximum.
This is important with regard to the world record chase for an obvious reason: the record can only go so high, and every time it's topped, it'll be that many points closer to the limit.
Hank Chien's reigning score of 1,090,400 is about 100,000 points away from the theoretical ceiling. Since Chien set the record in February 2011, two players—Dean Saglio and Jeff Willms— have achieved higher scores on the MAME platform.
Saglio's score of 1,153,000 is roughly 50,000 points away from the ceiling, and is going to be extremely difficult to beat, even by Saglio himself, who plays regularly and has yet to match it.
While the arcade machine record is nearly 60,000 points away from the MAME record, the chase (on either platform) could be sealed shut forever with just one more big game.
From the day that Billy Mitchell set his original record of 874,300 points in 1982 until Steve Wiebe first scored a million points in 2004, there was still plenty of room for the score to grow. But the situation ever since is akin to the final rounds of a game of musical chairs, and very soon it will be a matter of who nabs the single remaining seat when the music stops.
Four men have been champion, and there is only room for one, perhaps two, first-timers, assuming that Hank, Billy, Steve, or Tim don't boost the record to its final resting place... and at least two of them are actively trying.
On some level, the chase has already come to an end, in that skill is no longer the operative factor in getting the record. Skill must now be paired with good luck.
All top Donkey Kong players will agree that even when you are playing the game as well as it can be played, your fate ultimately comes down to what sort of fortune it decides to dish out, which will be anywhere on a range between "brutishly nasty" to "average" to "unusually favorable" to "slot machine jackpot."
Even million-point players don't get to a million every time, or anywhere close to every time. After he had played his first million-point game, Hank Chien described a kill screen game as a "one in ten" event, and a million point game as a "one in a hundred." Steve Wiebe asserts that his average game is only in the 600,000 point range.
Maximum skill combined with better-than-average luck will get you to a million points, but the degree of luck needed to achieve a score that would break Chien's or Saglio's current world records resides in a realm well beyond "unusually favorable." If you want 1.1 million points or more, skill alone simply won't cut it.
So, at this point in the race, we're no longer dealing with ability, but probability. And as the world record inches higher, approaching the ceiling, larger amounts of luck will be necessary to top it.
Eventually the score will be so close to its maximum that attempting to beat the world record will be more akin to entering a raffle than a contest of skill. There may be less than a dozen people holding tickets, but the odds will be heavily against them, both individually and as a collective.
Recommended reading: Steve Wiebe and the Future of Donkey Kong