I have heard that many people think that some Lubavitchers (obviously mistakenly) think that their rebbe is somehow more immortal than everyone else. Does anyone know if this is true and to the extent that it is, how many people we are talking about?
There is a small group of people who believe that the Lubvatcher Rebbe Menachem Mendel Schneerson zt"l is in fact God. For example, the front page of the website for a synagogue(?) in Mikwaukee reads, in part, "Yechi Elokeinu Moreinu V'Rabbeinu Melech HaMoshiach L'Olam Va'ed!"
As far as numbers are concerned, I do not believe any are readily available; however, Wikipedia does state, "Very few chassidim actually pronounce the sentence, 'The Rebbe is the Creator', though the number is not as negligible as one might imagine." As far as I can tell you from personal experience, I have never met anybody who actually believes that the Rebbe is indistinguishable from God.
Your question asks about people who believe that the Lubavitcher Rebbe is "more immortal than everyone else." This is a broader category than those who believe that he is tantamount to God. To some extent, anybody who believes that the Rebbe is the mashiach believes that he is "more immortal" than us regular people since, in order to fulfill the duties of the king mashiach, he would either have to be resurrected prior to everybody else or currently exist in some alive state-of-existence that we do not understand. Either one of those makes him "more immortal" than the rest of us in my opinion.
The group of Chabad chasidim who believe the Rebbe is mashiach is still a minority , but constitute a much larger group than those who believe he is God. I still don't have any numbers, but if you spend a decent amount of time visiting various Chabad houses, you are likely to meet a few of these people.