Why is Voldemort so obsessed with not dying?

Question

Before he had even left Hogwarts, Tom Riddle had developed a deeply unhealthy obsession with trying to obtain immortality. Not only did he consult numerous dark magic tomes but he also risked exposure of his villainous schemes by asking a teacher (Slughorn) for guidance on how to go about creating multiple horcruxes:

I, who have gone further than anybody along the path that leads to immortality [said Voldemort] You know my goal – to conquer death. And now, I was tested, and it appeared that one or more of my experiments had worked … for I had not been killed, though the curse should have done it
_{Harry Potter and the Goblet of Fire}

and

‘Well, Harry,’ said Dumbledore, ‘I am sure you understood the significance of what we just heard. At the same age as you are now, give or take a few months, Tom Riddle was doing all he could to find out how to make himself immortal.’
_{Harry Potter and the Half-Blood Prince}

So why is Riddle/Lord Voldemort so obsessed with not dying?

Was there a specific incident, perhaps something from his childhood or an event at Hogwarts or was it something else entirely?

Answer 1

I’m fairly sure this is never answered definitively, but I can make some guesses.

The death of his mother

Tom Riddle’s childhood was defined by death: specifically, the death of his mother. That’s how he came to be left at the orphanage. Listen to how he reacts to the news that he has magical abilities:

“My mother can’t have been magic, or she wouldn’t have died,” said Riddle, more to himself than Dumbledore.

— Half-Blood Prince, chapter 13 (The Secret Riddle)

With no knowledge of the magical world, he immediately assumes that wizards have the upper hand. Magical folk are stronger than muggles, and their power makes them immortal.

I think he saw his mother as weak for abandoning him, for succumbing to death and leaving him in an orphanage. This view was probably reinforced when he learnt the circumstances of her upbringing and his birth. (I can’t imagine Riddle was pleased to have an almost-Squib as a mother.)

This sets up the idea that death is a human weakness and frailty, and something to be bested. It also gives him a way to prove, once and for all, that he is better than his mother.

His desire to be different

We get another hint in the same passage: this time from Dumbledore.

“There he showed his contempt for anything that tied him to other people, anything that made him ordinary. Even then, he wished to be different, separate, notorious.”

— Half-Blood Prince, chapter 13 (The Secret Riddle)

Everything dies. (Well, almost everything.) There aren’t many ways to be more different than to not die. It would be the ultimate standout moment, something that would stand him apart from everyone, forever. It would assert his (self-believed) superiority over all living beings.

A desire for power

We know that Voldemort craves power over. JK Rowling has described him as a psychopath on multiple occasions, and we know that he was a bully in the orphanage. He didn’t just want to be different, he wanted to be better.

Voldemort saw death as a human weakness:

If Voldemort saw a boggart, what would it be?

Voldemort's fear is death, ignominious death. I mean, he regards death itself as ignominious. He thinks that it's a shameful human weakness, as you know. His worst fear is death, but how would a boggart show that? I'm not too sure. I did think about that because I knew you were going to ask me that.

— The Leaky Cauldron and Mugglenet interview JK Rowling, part 2 (July 2005)

So why does he think death is a weakness? Because from the perspective of the living, you can’t be any more powerless than dead. You can do nothing. Given his lust for power and influence, I think he was scared of that degree of helplessness. He would see it as unbearable to be unable to affect the world around him.

He doesn’t just fear it; he sees it as the worst possible fate. Quoting Dumbledore:

“Your failure to understand that there are things much worse than death has always been your greatest weakness—”

— Order of the Phoenix, chapter 36 (The Only One He Ever Feared)

There’s a complete dearth of any knowledge of death in Potterverse. There’s a vague notion that a soul survives after death, but in what form is unknown. I don’t think he could contemplate the prospect of being dead, and having to give up any influence on the physical world.

Answer 2

Building on alexwlchan's excellent answer, it's worth noting that we may have missed out the single most formative experience of Tom Riddle's early life:

His first murder/s

Voldemort's own fear of death (and keen interest in cheating death) seems to have immediately followed his decision to murder his family:

‘Meanwhile, in the village of Little Hangleton, a maid was running along the high street, screaming that there were three bodies lying in the drawing room of the big house: Tom Riddle Senior, and his mother and father.

It's pretty reasonable to assume that having learned about human frailty first-hand that he would see his own mortality in sharp relief. Note that by the time he speaks to Slughorn about Horcruxes, he's wearing the ring he stole as a continual reminder of his actions:

His right hand lay negligently upon the arm of his chair; with a jolt, Harry saw that he was wearing Marvolo’s gold and black ring; he had already killed his father.