I forget if there's a second one, I know Netflix says you can download on its ad but I don't have netflix to test. The one I know for sure that you can download and watch offline is Prime.
You MIGHT be able to download and watch offline with HBOgo. I know at one point I could. But that might also be one of the ones that took that feature away.
I also don't know about any of the disney/paramount/apple ones. I think I tested Prime, HBOgo, Crunchyroll (iirc this can do it but theyre always missing episodes its annoying), Youtube (requires connection) and hulu.
When I'm traveling, I download tv shows I think I want to watch and stick it in VLC on my iPad. The exception being movies from the apple TV app. I have had no problems renting and downloading the video through that.
Maybe they fixed it, but ever since I downloaded a bunch in the HBO app only to find it would NOT play once I was in the air, I just gotta do what I gotta do to be able to actually watch in flight.
Oh man, and the other thing apps will do, is if you're not careful, they'll download some awful low res version of the show by default. ugh.
I hadn't touched my ipad in 4-6 months I only use it on flights and I opened it up and had almost no apps. I might have actually had no apps but the apple apps if thats a thing. I couldn't open ANYTHING. Then I got to the airport for my next flight and was franctically running around hoping to find a guest wifi network..
That's disabled now.
Then there's the chromecast app. I want to watch an episode of survivor, a show with 45 seasons now. Apparently it just loads all the episode data for the entire show in one go, and hangs for a bit when opening the episode select page. This is one of their bigger shows I think.
It's just so lazy and bad.