The following mind-blowing theory comes from genius Reddit user Lampawarroo, whose careful research and masterful insight brings us what may be the greatest Star Wars theory ever conceived...
Here I will seek to establish that Jar Jar Binks, far from being simply the bumbling idiot he portrays himself as, is in fact a highly skilled force user in terms of martial ability and mind control.
Furthermore, I assert that he was not, as many people assume, just an unwitting political tool manipulated by Palpatine-- rather, he and Palpatine were likely in collaboration from the very beginning, and it's entirely possible that Palpatine was a subordinate underling to Binks throughout both trilogies.
And finally, given the above, I will conclude with an argument as to why I believe it is not only possible, but plausible that Jar Jar will make a profound impact on the upcoming movies, and what his role may be.
So first, let's establish Jar Jar as a skilled warrior. While this does not in itself necessitate a connection with the Physical Force, it's highly suggestive in the Star Wars universe-- very rarely do we see "normal" characters exhibiting extraordinary stuntwork or physical feats unless they are Jedi, Sith, or at least force sensitives.
So here's Jar Jar nonchalantly executing a standing 20 foot twisting somersault:
Now, taken out of context, if you were watching a Star Wars movie and saw a character casually execute this maneuver, you'd probably assume it was a Jedi. In the context of Jar Jar, though, we don't... because elsewhere he so thoroughly convinces us that he's nothing more than a harmless dunce with his inane dialogue and cowardly-lion act.
He also manages to convince us that he's a bumbling oaf in the midst of pitched battle... even though he's always incredibly, amazingly successful. Whether single-handedly taking down a battledroid tank, or unleashing a barrage of boombas on their front lines, or precisely targeting multiple enemies with a blaster tangled around his ankle (!!!), we simply roll our eyes and attribute it to dumb "luck."
This is one of the main reasons we as an audience hate Jar Jar so thoroughly; he breaks the fourth wall, he shatters our suspension of disbelief, because we know that no one is really that lucky. We dismiss it as a lame, cliched trope-- the silly pathetic oaf who always seems to inadvertently save the day.
I posit that, instead, this is a deliberate facade on the part of Jar Jar as a character, and on the part of the writers and animators. As we know, the Jedi themselves are inspired by Shaolin Monks, and there's a particular kung fu discipline that Jar Jar's physicality is purposefully modeled upon which allows him to appear goofy and uncoordinated even as he lays waste to his enemies; namely, Zui Quan, or "Drunken Fist wushu".
This discipline seeks to imitate the "sloshing," seemingly random foibles of a drunkard, but in reality the staggering and stumbling is the use of bodily momentum, deception, and unpredictability intended to lure and confuse opponents.
Let's take a look at Jar Jar displaying some drunken fist techniques.