Spring, March in particular, always seems to be the time when I discover new shows. I might watch a new show at any time during the year, but it's always at this time that I find the ones I get absorbed into. This year it's been Doctor Who, last year it was Once Upon a Time, the year before it was Primeval, then Harry Potter (which isn't a TV show, but watching all the movies as a sequence is sort of like watching a show), and in 2010 it was I Dream of Jeannie.
While I was watching Doctor Who, I found myself comparing certain elements to I Dream of Jeannie. All of this varies greatly depending on which doctor or which companion is on the screen at the moment. But let me go through the general idea.
Jeannie lives in and sometimes even travels in her bottle, which has become a symbol of her character and of the show, just like the TARDIS. (For the record, I prefer the bottle: though the TARDIS is great, it's just so blue.) Jeannie is magical; though the Doctor is not, he does have magical-like characteristics because he is "clever," uses science from another world, and regenerates every so often. Likewise, they're both off-kilter from the rest of (our) society. They dress different, they talk different, they get bored by everyday life.
Jeannie has Tony, whom she is always trying to awe and amaze with gifts and travel. The Doctor acts similarly around his companions ("No one is around to see me being clever"); he likes to show off the universe to them. The big difference, of course, is that Jeannie is trying to woo Major Nelson, while any extra feelings that come about between the Doctor and one of his companions are never planned--or entirely wanted. But when a relationship does develop, in either show, it is very sweet--despite the fact that it, in a way, goes against the entire show. People still complain that Jeannie and Major Nelson were married in the last season (I don't), and the Doctor and Rose could only be together with the plot-invention of a clone and a parallel universe.
Both shows, I think, win in their mixture of the regular world and the sci-fi/fantasy world. We see regular things and places, but we also adventure into history and either magic or space. The horizons grow, and the characters are constantly trying to keep a fix on reality and to keep everything organized.