Single celled organism and immortality
I don't believe this question is an exact duplicate of the question at Do bacteria die of old age?, however all of the comments and answers provided there are relevant, and you should read them before continuing to read this answer because I won't repeat what's stated there.

In addition to bacteria and yeast which produce certain "rejuvinated" cells that can divide indefinitely while other cells become senescent or "aged", all multicellular organisms actually employ a remarkably similar strategy. Under certain culture conditions, embryonic stem cells can propagate indefinitely (1).

During early embryogenesis they divide "symmetrically" to the extent that all of the daughter cells are fully undifferentiated until after gastrulation (2), but there is a certain degree of asymmetry that defines the axis along which gastrulation occurs prior to gastrulation (3). This asymmetry is mediated by a large number of signaling pathways (3), some of which become asymmetric during the very fist cell division of the embryo. Adult stem cells also divide asymmetrically primarily as a precursor to differentiation, but this is thought to also be involved in the asymmetric distribution of damaged components to their more differentiated daughter cells (4), as occurs in yeast and bacteria. Currently there is less research on this in adult or embryonic human stem cells than in bacteria or yeast, but it stands to reason that our embryonic cell line does the same thing beginning at its first division, and our germ line is immortal (we're no "older" at birth than our great-great-great-great-great-great-grandfathers) for the same reasons that unicellular cells are immortal.

References:

(1) https://en.wikipedia.org/wiki/Embryonic_stem_cell#Propagation

(2) https://en.wikipedia.org/wiki/Embryonic_stem_cell#Pluripotent

(3) https://en.wikipedia.org/wiki/Gastrulation#Loss_of_symmetry

(4) http://www.ncbi.nlm.nih.gov/pubmed/25966295