I can't figure out how to hide all content from public. I removed all privileges from group Anonymous and set it the default group. Also I created a new group Public without any privilege, and made it the default group. Nothing seems to work.

What I want is a frontpage, wiki or article with some info about the site and how to apply for a account, and log in boxes.
Preferably no top menu is visible. So far I only managed to get rid of the My... menu item.

Look in Admin > Themes > Module Layout / Column Settings. There you can allow/disallow groups to see the different modules. "Modules" are the boxes on the sides, the menu, the top bar, the footer etc.

You mean Administration > Themes > Manage Layout \ Modules?

Then the procedure I should follow is to assign Site Default to group Public only, set the Users / Login Box for some left or right column, and assign a Wiki page with instructions how to apply for an account as homepage of this group?

Sorry, I meant "module settings". The address is /kernel/admin/index.php?page=layout_overview

Basically, set the field "groups" for all modules to "admin" except the login module, similiar to this:
http://www.iimmgg.com/image/57aa945b14d5f0cbbe20ab4642708750

The module login you wanted to keep, so that module would stay on "anon".

Then, in Admin > Users > Permissions, you take all the permissions away from the group Anonymous like you did. "View" for any package, "edit" for any package and so on. Leave only the permission to view wiki pages. In Admin > Wiki > Settings are some wiki centric settings as well like printing, I think. This should be it.

If you know Smarty etc. a little bit, another way could be to hack bitweaver/kernel/bitweaver.tpl. Copy it to bitweaver/themes/yourtheme/kernel/bitweaver.tpl and wrap everything in there into {if $gBitUser->isAdmin()} except your info and an additional { include file="bitpackage:users/login.tpl" }. This might work.

In 2.6.0 it is "Configure layout options".

Thanks for the picture. I realize now that the modules are expandable...

The result of your suggestion is that I get the "Login Box" and the "Login Page" and a login link in the upper right corner. If I set a Wiki page as "Home Page" of groups "Administrator" and/or "Anonymous", it is not shown.
I added a user with default group Registered and set the field "Groups" to both "Administrator" and "Registered". This works.
But how can I get rid of two of the three login references and also get the wiki front page visible?

An alternative way I followed was to remove all privileges, but unfortunately I can't get rid of the "Boards" menu item.

I'm not a php and/or smarty developer at all, nor do I have the brains to learn it quickly.

Removing all permissions from anonymous should cause almost every page to be blocked. What urls are you able to access that seemingly should not be accessible?

For instance mysite/wiki/index.php.
But it looks like I've messed up the site completely. Only after a fresh install (with 2.7 rc1) I managed to implement advice given by Laetzer.
Thanks all!

Looks like Leatzer's solution doesn't work for me. All wiki pages are visible for Anonymous with direct links.
So it looks like I have to edit login.php with introductory info?

The simple way of doing this IS probably just to create your own login page, if that is all you need for 'anonymous' access.
An alternative is protector service which I created simply to allow some content to be visible anonymously, but most to be hidden until one logs in. ProtectorService

This problem is so simple to solve I'm confused how this thread has gone this far off the rails. Steps to do what you want:

1) install wiki package

2) set permission to view (and any other wiki permissions) to Editors or higher (this assumes you dont want common registered users browsing your pages)

3) create the page you DO want public and set its custom content permission for viewing to anonymous group. there is link for setting custom content permissions in the icon menu.

you are done, all your pages except the ones with the custom content permissions will be hidden from anonymous and normal registered users.