A friend was recently telling me she had heard she needed to have letters written to carry with her justifying her need to carry her prescription medications while traveling internationally. (South Korea & Japan)

I am not sure what medications she was referring to, I assume controlled meds.'

Does anyone know of any countries that would require such a letter for HRT related meds?

Estradiol, progesterone, spironolactone, etc

Most countries carrying the medication in the original container is enough but some countries have laws agains drugs that are commonly used drugs in other countries. As such, it's best to learn the restrictions of the country you wish to visit or you could end up doing serious jail time. Also be ware that planes often make unexpected landings so you could end up with a layover in an unexpected airport with  a night layover. If you receive your  luggage, it may be inspected again presenting a possible problem so don't take more medication than you can comfortably dispose of.