For some reason, I thought of the Twilight Zone episode where the guy sells his soul to the devil and for immortality, then kills his wife to try and tantalize the police into giving him the death penalty. Of course, then the guy's lawyer get's him to life in prison, to which he says "Besides, you've only got like what, 30-40 more years to live? This won't be that bad."
That's one of the main reasons immortality would be an issue. The events in which the average human would die from starvation or suffocation or just life in a box, there'd be no escape. The worst thing that could happen to an immortal is developing FOP after you become an immortal. That would be a fate worse than death.
If you don't know what FOP is, it's when your ligaments become damaged, and instead of regrowing like normal ligaments, they ossify, which means they regenerate with bones. So, basically, if you have FOP, you're pretty much a statue until you die of starvation. The only things you could move would be your tongue, eyelids, and possibly lips.
I'm sorry for giving you all nightmares.