1. Gym Log+
My app. There is no account, no server of ours, and since version 3.15 the app makes no network requests of its own at all.
Open it and log; that's the whole onboarding. Your routines, history, body log and progress photos sync between iPhone and iPad through your own private iCloud database, which is encrypted and scoped to your Apple Account — we have no access to it. The Apple Watch keeps its own copy and runs whole workouts on its own. Export is free and complete: a CSV and a full-fidelity backup file, any time, no subscription needed to get your data out. If you ever want it gone, delete the app and remove the iCloud data from your own settings; there is no "delete my account" request to send anyone, because there is nothing to delete on our side.
Pick it if you want sync without a sign-up, and a logger that is yours outright ($79.99 lifetime, or $4.99/month). Skip it if you need Android or a web app — the same no-server design that makes the promise true is what rules those out.