I need a specific MP3 player. Any recommendations

If it is going to be plugged in all the time, why bother with an MP3 player?

Basically you just need a cheap system that will play audio continuously.

I am familiar with Brightsign which makes a system that will loop video or audio. I think the current low end version is the Brightsign HD120. It's pricey, $250, but it will work for a long time and loops without any problems. You might be able to find a used one or an earlier version, the HD110, for less. It does a lot more than what you want, but it is reliable and has a large user base.

If you want to go cheaper, just google "media player". There are a lot of them that are under $50. Whether they will loop is another question.

My $40 DVD player will play a list of mp3s, on a pen drive or burned on a disk, indefinitely. That list can have one or more entries.