Avoiding Death

Do you mean biologically immortal cells that do not suffer cellular senescence and that are not subject to the Hayflick limit and that can keep dividing without the shortening telomere limit that most cells have? Or something more along the lines of asexual planarian reproduction where a multi-cellular organism separates itself in order to reproduce asexually?

I am no biologist and am probably am not the guy with the answer you are seeking either way but I'm not sure your question is clear enough (though it may well be to those who are).

I think some animals (or other organisms) divide into two. Therefore the "parent" never dies but its cells continue in its two " daughter" organisms.

In every cellular division there is always the chance of error/variation/mutation, so even if an organism managed a precise division, the chances of it continuing to do so repeatedly are slim, so it can't continue forever.

On the other hand, you could almost make the argument that DNA itself never dies. It's been around for almost 4 billion years, replicating and changing. The original strand is long gone, but its descendants swarm the planet.

Take a look at New York Times Magazine Dec. 2, 2o12. Cover story about the "immortal jellyfish".

Yes. Reading that article gave me the idea for my able2know post. I thought possibly some higher animals reproduce that way.

With animals exhibiting advanced characteristics, understanding aging gets abstruse.
In some animals, a high correlation was found between lifespan and rate of telomere deterioration, essentially from observation that the possible limit of cell divisions are limited by telomeres.
So far, the mystery of aging is somewhat a longways before being fully cracked.
Another variable as crucial in the aging process as telomere deterioration when DNA replicates, are mitochondrial DNA mutation and dysfunction over time from the process of making energy (redox reactions), in some cases at a rate faster than DNA.
Some studies have also found genes related to aging.
Alter all such mechanisms (and possibly more to discover with further research) and there is some hope in the prospect of avoiding death a little longer.
How dismal that evolution limited an organism's clock with a deluge of mechanisms at cellular and molecular level. I suppose nature being a ruthless boss, we have our work cut out for us.There is senescence for you.