Very interesting. My main concern is that such systems should suit their owners very well. But, of course, everyone has their own preferences regarding task management, time tracking, app ecosystems, usage patterns etc. Consequently, it is very hard to make a single open source project that fits everyone.
Me and my friend were working on a very similar project for the last 10 months, and we still hesitate to publish it (also, it is written in Scala, thus limiting potential audience). Below, I provide more information about our work, some insights discovered during development, and future plans. I hope it could help other projects too.
The project is focused on outsourcing memory, providing smooth, most of the time automatic, time-tracking, being hackable and keeping personal data private.
Key features needed for that are different integrations (we use JIRA, Trello, Hubstaff, Trakt.tv, Apple Health, sleepasandroid, Beddit); constant location tracking; convenient input (currently, Telegram bot with NLP capabilities, suggestions and autocompletion).
Eventually, it turned out, that chatbot interface works like a charm for actions like forwarding links to read them later, managing tasks in JIRA (when chatbot sends you updates, and then you can use your messenger to comment on or manage tickets; also works well with time-tracking), aggregating notifications.
We also have some analytics: daily briefs, time tracking summaries, metric-based reports. Another features include global search, flashbacks, GPX track export (to use with Lightroom, for example), history-based notifications. Future plans include: finance tracking, delivery tracking, ReSpeaker integration, newsfeed aggregation.