You can
download YouTube video, then replay it or several videos from the playlist over and over again.
For example, drag the downloaded video onto VLC player. Then hit the "Show playlist" button. Then hit the Loop button and choose "loop one" for replaying your video.
You can also add multiple videos to the playlist and then choose "loop all" to replay your playlist.
You can also use
TubeReplay, a free web-site allowing you to watch your favorite YouTube video over and over again without pressing the Replay. Just copy the URL of the YouTube video, paste it in TubeReplay and hit the Replay button.