Rant update:
The mobile emitter highlighted one of the biggest problems with the EMH as a character.
See, as soon as they had a mobile emitter for him, people started to notice that, if the mobile emitter has enough memory to hold his program, and the sickbay holosuite has enough memory to hold his program, there is no reason at all why he can't be in both places at once.
And there really isn't a reason why he couldn't do that: two different Doctors. They'd be individuals: there is no reason why one would know something that was happening to the other, but they could merge their memories periodically to keep up-to-date.
Once you've realized that, and how little physical space the amount of memory required to hold the Doctor represents, you realize that there could probably be 2 Doctors simultaneously in sickbay. The Doctor doesn't need an assistant: he can
be his assistant. And his own Nurse. He can work 3 shifts and still go hang out on the holodeck.
Oi! The holodeck! Even if the sickbay holosuite has a sharp limit on how many characters it can hold in memory, the holodeck can do a handful at once. Granted, at any given time some of the "people" in Fair Haven were likely just moving images, but like a dozen of them at once were real, interactive fully-realized holoimages. So the Doctor could use a holodeck as a small hospital, with a Dozen doctors who were all Doctors.
A bit expensive from the effects-budget end of things, but a million laughs.
Think of it: a recurring gag where, while one person is being examined by the Doctor in sickbay, another Doctor walks by carrying a tray, and they talk briefly.
The Doctor is a computer program. Like Moriarty, he is an apparently sentient computer program, but still just a program. He can be "cloned" by copying his program into any system with enough memory to hold it: if it can
run the program, so much the better. As such, he is pretty much immune to permanent death, and only able to manefest in the physical world if there are holoemitters.
I personally like the idea that the holoemitters in sickbay are special: that the Doctor's ability to function as a doctor is tied to the ability of those special emitters to create more sensitive force fields (or whatever), and that when visiting the holodeck the Doctor loses much of his sense of touch, for example. But still, the guy's unique quality is his very lack of uniqueness.
