If you mean transparent polymer as "clear polymer"", I recommend you PDMS as a transparent and biocompatible polymer with ,more than 100% of stretchability. You can check my article for more information:

https://www.researchgate.net/publication/261767953_Highly_Stretchable_and_Sensitive_Strain_Sensor_Based_on_Silver_Nanowire-Elastomer_Nanocomposite?ev=prf_pub

Article Highly Stretchable and Sensitive Strain Sensor Based on Silv...

with just 2-3% elasticity you can sub 100 micron pet sheets (40-60 m will do). 

Thanks everyone,

The PDMS sounds great, but I'll start with the materials that are easy to get hold of.

I guess, by definition, any of the Poly's should regain their original dimension if they are thin enough.

Thanks again

there are many flexible substrates such as PE, PTFE, PP, PC, and so on.

Best wishes!