I could use the opposite:
A TODO-app which has no synchronization capabilities so I don't need to worry about leakage of private data.
Ideally it would use a very simple plaintext file format for its database so it is future proof, e.g. http://todotxt.org/
The TODO.txt format seems very nice but the Android apps listed on that site are either not fully developed or unmaintained, and one even looks like it's not open source.
Simpletask seems like the most advanced implementation, but it got kicked out of Google Play for the usual Google shenanigans, so the author says on a sticky GitHub issue that he has lost motivation: https://github.com/mpcjanssen/simpletask-android/issues/1110
I've been using https://github.com/vimoutliner/vimoutliner for about a decade, everywhere I work. Just text files with simple Ascii checkboxes.