When carrying medicine on a road trip, I have sat it (in a ziplock bag) in the ice bucket overnight and packed ice in the cooler in the morning for the next day's drive. There's no such thing as a usable mini fridge anymore, they're all mini bars fully packed with pricey items.
Right?
I've never seen a hotel fridge with anything in it. Hell I'm actually staying at a hotel right now that's pretty decent and was reserved by my company. Smallest mini fridge I've seen, but it's empty and could certainly fit some meds.
I guess I go to nicer ones. I haven't seen alcohol in years (illegal in my state anyway), but most at least have sodas or water bottles. I never know what's complementary, so I just avoid it.