Yes, you can start writing with one-click; there's no sign up and no install required. This is so you can try Overleaf with the minimum of fuss, and if it's not for you, you've made no commitment. Click here to create a new paper with the default Overleaf template.
We hope you will like it though, and if you do, the option of signing up gives you extra control over your work, such as being able to manage all your documents from one place, upload extra files to your projects, and to save & restore different versions of the same document.
If you're new to LaTeX and would like to learn more, we have a set of video tutorials to help you get started.