listenonrepeat.com was up for years until last month, when it mysteriously went offline. I haven’t found any sites like it. I could just paste in a youtube link and have it play right away, and choose the start and end times for looping, and it had a count for how many times you played the song, as well viewing history (with start & end times saved) so you could easily listen to previous songs.
Found somewhat recently an Android program, Musicolet, which does count how many times you played a given music file. It’s on Google Play and as an APK through the devs’ site.