I absolutely love to look back at my whole year and see how many games I completed/played. It brings back memories, and it feels almost like scrolling through old family photos. But what with games that cannot be “completed”? Like roguelikes, MMOs, games with endless grind and replayability.
Leave it in the “currently playing” section forever, add it to all shelves with years that I played this game, Add it to a completely separate category, or perhaps you guys have some other, different solution?
Most of those games have a campaign or endboss. Beating those counts as completing them for me. For multiplayer games you can just come up with your own standard of having “completed” the game (e.g. playing it for 100 hours).
Usually, I add games to the “Played” shelf after I stop playing them for the first time, regardless of any content I’m yet to experience. If you follow this path, you can have a separate “Completed” shelf if you want to be able to differentiate between games you’re “properly” done with (either 100% completion or a traditional end to the main story) and the ones you’ll never be out of stuff to do in. I prefer to keep the “currently playing” for games that I’m actually touching over the last few weeks.
The criteria for completion is obviously something totally personal, hence why I recommend basing it on whether you feel done with a game or not.