[simonsquiff]

Thanks - germ cells was the key term I was missing, this section on the germline on Wikipedia answers my original question:

Germline cells are immortal, in the sense that they have reproduced indefinitely since the beginning of life. This is largely due to the activity of the enzyme known as telomerase. This enzyme extends the telomeres of the chromosome, preventing chromosome fusions and other negative effects of shortened telomeres. Most somatic cells, by comparison, can only divide around 30-50 times according to the Hayflick limit. Certain somatic cells, known as stem cells, also express telomerase and are potentially immortal.

[Houshalter]

So then why can't we just inject ourselves with that enzyme and be immortal?