A good premise is hard to find these days, and John Doe's premise is one of the best. Having the geeky trait of knowing everything but himself allows anyone to put their own background in, if only for the moment, making Doe someone with whom we all can easily identify. It also gives the writers carte blanche and an arc story replete with possibilities to develop over the seasons.
The trade off with a carte blanche is that if you have no idea who John Doe is (and are reluctant to find out as a writer) you start getting episodes that are shallow. Take this 'Mourner' episode. A serial killer learns of John Doe's amazing abilities and nonexistent past and decides to play deadly games testing John Does abilities. Sounds good, but when you watch the episodes there is no heart in it. It doesn't have that magic where the hero meets his nemesis. A nemesis is someone who is your match, who is essentially you, but your opposite. The writers should have slowly revealed to Doe that the Mourner knew what he did - basically everything. Instead we get a nemesis that's a diluted version of the countless madmen we've encountered of the years in B movies and TV shows. Maybe I'm jaded; with serial killers abound in shows like The Mentalist, Psych, and Law and Order: SVU, maybe creating The Mourner for John Doe seems to be, no pun intended, a copy-cat. In fact, I believe it's pointless to introduce a nemesis at this point in the story because we are still learning who John Doe is as a person. We don't have a good sense of his boundaries. Is he willing to kill? How does he change when under the pressure? Doe's encounter with The Mourner was the first time we see him sweat and it was oddly out of character for him.
That's the problem with carte blanche. We have no idea who John Doe is.