Reddit reviews The First Fifteen Lives of Harry August

The Goblin Emperor by Katherine Addison.

The First Fifteen Lives of Harry August by Claire North.

Wool by Hugh Howey.

Half a King by Joe Abercrombie.

Read through the "Look Inside" previews to see if any of these is right for you.

If you love this stuff then ill also recommend [The First Fifteen Lives of Harry August] (https://www.amazon.com/First-Fifteen-Lives-Harry-August-ebook/dp/B00ECE9OD4#nav-subnav)

Same general idea, a very small portion of the population relives their lives from birth each time they die.

I'd repeat indefinitely. Memorize a few lottery numbers and when to play them. Memorize a few key stocks and when to invest. The world is a big place; even with eternity on my side doubtless I could see it all. Life would be like a giant role-playing game that I play over and over; playing the same physical character each time but different role on each pass. Rich media tycoon; homeless (but secretly rich) drifter; swinging bachelor. The only roles that would I would find difficult to play, at least over and over, would be of Family Man, creating children who wouldn't exist the next time around, winking out of existence except in my memories.

Edit: Reminds me strongly of the book "The first fifteen lives of Harry August"

This wasn't a short story, but it has some similarities to what you described: The First Fifteen Lives of Harry August.

https://www.amazon.com/First-Fifteen-Lives-Harry-August-ebook/dp/B00ECE9OD4

He sorta time travels, but not from any machine. When he dies he starts his life back at the time he was born with his previous memories. Other people have that sorta ability too. There is a sorta police interrogation scene where he describes stuff like this to them. Probably not it though.

Here's a book for you: https://www.amazon.com/dp/B00ECE9OD4/

For interesting time travel concepts that are like the Butterfly Effect or Inception....then:

1. Replay
   
2. Dark Matter
   
3. The First Fifteen Lives of Harry August
   
4. Lost Futures
   
5. The End of Eternity

The first 15 lives of harry August. http://www.amazon.com/gp/aw/d/B00ECE9OD4

The Goblin Emperor by Katherine Addison.

The First Fifteen Lives of Harry August by Claire North.

If I remember correctly "The First Fifteen Lives of Harry August" is without dialogue and I really liked the story.

Find out the answer to your hypothetical here:

https://smile.amazon.com/First-Fifteen-Lives-Harry-August-ebook/dp/B00ECE9OD4/ref=sr_1_1?s=digital-text&ie=UTF8&qid=1474292285&sr=1-1&keywords=harry+august#nav-subnav

It's about a guy who keeps living his life over and over again, from birth to old age.

The First Fifteen Lives of Harry August by Claire North.

The City and the City by China Mieville.

Angelmaker by Nick Harkaway.