Mobile-friendly, minimal, private, and open-source. Works great on your phone as an add-to-homescreen web app, and there is no lock-in with pre-paid hosting.
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Todoist - Todoist is a to-do list that helps you get organized, at work and in life.
Evernote - Bring your life's work together in one digital workspace. Evernote is the place to collect inspirational ideas, write meaningful words, and move your important projects forward.
Remember The Milk - Remember The Milk is a task and time management application for mobile devices.
Joplin - Joplin is a free, open source note taking and to-do application, which can handle a large number of notes organised into notebooks. The notes are searchable, tagged and modified either from the applications directly or from your own text editor.
Google Tasks - to-do list app that integrates with Gmail and Google Calendar
Google Keep - Capture notes, share them with others, and access them from your computer, phone or tablet. Free with a Google account.