Would you want that? Would you choose to accept it?
What if you wound up with immortality, and found out too late that it was more a curse than it was a blessing? Would you then spend the rest of your existence trying to find a way to be mortal again?
Three thoughts on the subject: the first is that immortality is possible but it's not something meant for this material realm. 1st Corinthians 15:50 teaches us that "Flesh and blood cannot inherit the kingdom of God". I'm now taking this to mean that physical immortality in the state of matter as we know it is something fully counter to God's intended order for the cosmos and concordantly, the pursuit of such a condition would be abomination. What then would be the ramifications of actually achieving this very thing?
The second is that the moment a person might become immortal, he or she would cease to be a part of the human narrative and experience, and would become something completely unnatural and alien.
And of course the most damning thing of all about immortality: it would probably happen in such a way that you could not let others share the "gift", and if there are a lot of people in your life then you would be condemned to watch them all pass away, to whatever is waiting on the other side, while you have to remain here and never be able to join them. Your own life without real fear of having to ever taste death, weighed against ultimately being alone and never seeing your loved ones again...
Would you choose to live for yourself in this carnal plane until the end of the Earth, or would you choose to die in the course of time and so ultimately move on, to where the ones you care about are waiting?