User login experience for web app

I have designed what I believe to be the simplest possible user experience for a website. There are only 4 buttons in total on the whole page and the look is very clean.

But there is nowhere to put the logout button. The way I have it setup, if you try to do an action that requires login a modal will pop-up for you to login/signup. Once that's done the modal disappears and the action executes.

I don't see a good place to put a logout button and I really don't want to mess the UI by placing menus / etc. I am considering two options:

1) Make the browser lose the session when the window / tab is closed. This has the disadvantage of requiring users to login the next time they open the app. I am not sure I am willing to accept such experience as it may discourage them from performing authenticated actions (which I want them to perform).

2) Leave the users logged in forever, with no way to log out other than clearing cookies.

Would you consider 2) an acceptable experience? It is not really a confidential app or anything.