I like to listen to the archives of the BBC podcast "In our time", which contains more than 750 episodes. But I can't see the all the older episodes with Apple's app, which has a 300 episode limit. Is there any iOS app that doesn't have an episode limit and so will let me see the entire episode list and download any episode from the archive?
300 episode limit is due to podcast host, not podcast app.
I'm a long time listener to "In Our Time" as well. Instead of subscribing to the main "In Our Time" podcast, with the 300 episode limit, you can subscribe to the five "In Our Time" podcasts of History, Philosophy, Science, Culture and Religion. As of today the episode counts for these are:-
In Our Time: Culture 206 In Our Time: History 265 In Our Time: Philosophy 126 In Our Time: Religion 103 In Our Time: Science 243 Total 943
According to wikipedia, "In Our Time" is currently (24.6.2018) at 808 episodes, so there must be some duplicates in those five podcasts.
Extremely large playlists now only show the most/least-recent 500 episodes to improve app performance for users with very high subscription counts.
Feed, Not App, Limit
The number of available episodes may be limited by the BBC, and not the podcast application. The feed used by the podcast application is provided by the BBC and does not contain every episode.
Counting Episodes in the Feed
You can prove this by downloading the BBC In Our Time podcast RSS feed. Use the following command in Terminal.app to download the complete feed:
curl http://podcasts.files.bbci.co.uk/b006qykl.rss > ~/Desktop/in-our-time.rss
Next count the number of file enclosures mentioned in the feed:
grep -c '<enclosure ur' ~/Desktop/in-our-time.rss
The result, as of answering, is
244 file enclosures.