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MegaCast - Chromecast player?
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Popularity Score: 5
This app was mentioned in 5 comments, with an average of 1.00 upvote
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megacast is what I've used. It converts as it streams. Not perfect, but much better than dicking around converting every damn video into three formats
This isn't what Plex is for. The Plex app is for streaming from a Plex Media Server hosted elsewhere (i.e. running it on your desktop, NAS or proper server) to your phone, Chromecast or remote controlling other Plex media players.
For your case, I found this app by typing "chromecast transcode" into Google Play. I've never tried it but it claims to be able to transcode and has a high review score.
Alternatively, you can install an ffmpeg app and transcode the audio/remux directly on the phone and then cast that output.
(Curious, why is the media on the phone instead of a computer or something?)
megacast searched for mp3 player chromecast