For a long time my partner and I used a shared Google Doc for a grocery list - it was great that it was real time and all, but kind of fiddly - hard to click the little checkboxes, and Google Docs is slow on a phone. So I made up a simple grocery list webapp.
I'm sure there's a billion of these out there, but I had some particular preferences:
* very easy to check off things on phone
* works on this ANCIENT iPad I now have set up as a permanent screen in the kitchen
* no chance someone is going to start trying to charge me a subscription
* items to get are sorted by the section of the store they're in
* you can add a comment to an item
* recently checked off items (in the last 60 minutes - arbitrary but effective) show up as crossed off items below
* checked off and older items are sorted alphabetically
In the interest of UI simplicity there are a few weirdnesses: no actual "checkboxes", you can only edit the section or note for an active item, and only delete an archived item, but honestly I think got the UI just right for our needs.