In fact, I don't believe that one can experience it at all. Imagine the procedure (for a conscious person): your brain is connected to a computer interface and a copy of your mind is taken. Great. Now there is a digital copy of your mind. So what? You still get to die.
Consciousness is mortally bound to the physical body, and will die along with it regardless of how many mental copies are made.
I much prefer the idea of human metamorphosis. I want to inject nanorobots into my body which will transform it gradually, cell by cell into an improved synthetic one. In this way, immortality and superhumanity can be achieved without loss of continuity. I imagine the experience would be one of waking up a little better (stronger, smarter, etc.) every morning until the cells are all upgraded.
I don't read enough these days because of chronic eyestrain, but if you are aware of anything dealing with this concept of human metamorphosis, fiction or otherwise, I would appreciate a link. It's something I would like to read about (if it's even been dealt with), someday when I have time (and don't have a headache).