Word meaning "greedy desire."
Solution 1:
The standard word for greedy desire is lust. By itself, the assumed meaning is sexual, but it is often used in compound constructions such as blood-lust, power-lust and money-lust. I've never heard immortality-lust, however, as @EdwinAshworth mentioned, you could use the longer phrase lust for immortality.
The phrase lust for life is more common, but it doesn't mean immortality in particular. Still, it might be the best fit for your desired sentence in particular.
Many had died in their attempt to be immortal. Kings, emperors, merchants, scientists —the lust for life doesn't discriminate.
Compare also the term lusty which you might expect to have sexual connotations, but which actually means healthy and strong; full of vigor.
Solution 2:
Perhaps the adjective acquisitive used as a noun
Excessively interested in acquiring money or material things.
Similarly, avaricious
Having or showing an extreme greed for wealth or material gain
Oxford Dictionaries Online
Solution 3:
There is covetousness, from covetous, which Merriam-Webster defines as
feeling or showing a very strong desire for something that you do not have and especially for something that belongs to someone else
All the first part of the definition requires is that the covetous one does not have the desideratum. So one may covet immortality even if no one else has achieved it or if it is unobtainable, as this use from Henry R. Cleveland's Felton's Homer suggests:
The idea of man, to which we have already alluded, conduced much to this. His very existence was tragedy; ... coveting immortality [emphasis added], yet certain of death, ...
But it may also be used if someone else has achieved immortality, either literally (perhaps through some device like the philosopher's stone) or figuratively through an enduring legacy.
In this case, envy may also be used, according to Merriam-Webster:
painful or resentful awareness of an advantage enjoyed by another joined with a desire to possess the same advantage