Star Wars is a part of culture that I hold very close to my heart (Don't all swoon at once, ladies). I would say that the mythology behind the series is the closest cultural equivalent to a modern day Arthurian Legend, setting up tales of morality in an expansive and pliable universe. Now that I've said all that smart sounding shit, it's time for me to argue something really stupid: Jar Jar Binks is the most complex character in the entire Star Wars universe.
Let me break it down to you on a movie by movie basis...
Star Wars: Episode I-The Phantom Menace
In Binks' first appearance, we're introduced to a sad character who has recently been exiled from his own culture. Almost immediately, he's taken off-planet for the first time and his entire world widens. Throughout this terrifying and alien experience, J.J. approaches it all with a Falstaffian jollity.
That very happy-go-lucky attitude is what draws the trilogy's lead characters to him. That will become very important later on. Eventually, Jar Jar's exploration of other cultures allows him to save the Gungan race from extinction at the hands of the Trade Federation. This allows him to be re-accepted by his culture, which he rejects in favor of the Universe and possibilities that are now open to him.
Star Wars: Episode II-Attack of the Clones
Jar Jar has been kicking around with Padme Amadala for a while now and, when she has to go into hiding, is entrusted to sit in as Senator for one of the most politically significant planets in the entire galaxy.
This fresh-faced, naive, innocent Gungan is almost instantly manipulated by Chancellor Palpatine in a power-play. For the first time, we are shown Jar Jar's jester-like demeanor and optimism working against him.
Star Wars: Episode III-Revenge of the Sith
This is the film in the trilogy where Jar Jar has the smallest role and that is very intentional. In his few, brief appearances, Jar Jar Binks is shown to have become a silent and sullen character.
Look at those eyes. Those are the eyes of a sad, sad soul. Jar Jar was given everything he ever wanted in life, and that ultimately ruined the one thing that made him special in the first place.
...Or he's just a silly, stupid annoying character. Who knows?