Asexual reproduction and Telomeres
Here is a study of planarian worms, which are immortal in asexual reproduction and mortal in sexual reproduction. 

Hydras also become mortal after they reproduce sexually.

Relevant to your question: Cells within planarian worm differ in expression of telomerase active subunit depending on body part. Immortal (asexual) worms have more expression in the area around the pharynx, those cells there are "the proliferative stem cells". There is a nice figure in the article.

I assume it is similar for hydra, that outer body coat has less expression than the cells inside. “Hydra is a bag of stem cells”

(Less relevant (a version of previous version of my answer. But does not answer the "every cell" inquiry well): In planarian worms "We demonstrate that this difference is reflected in the expression and alternate splicing of the protein subunit of the telomerase enzyme." The mentioned telomerase subunit was 1. comparatively more transcribed in immortal (asexual) worm and 2. within the resultant trancripts, there were possibilities of functional and non-functional isoforms due to alternate splicing. Immortal worms had higher ratio of functional isoforms. Summary: immortals have more transcripts and higher proportion of functional versions of them.)