Is there any way to give all users access to one blog in a multisite network without using a plugin?

You could filter get_blogs_of_user and add this particular blog to the returned list. Pseudo-code, not tested:

add_filter( 'get_blogs_of_user', 'add_special_blog', 10, 3 );

function add_special_blog( $blogs, $user_id, $all )
{
    $new_blog = get_blog_details( $special_blog_id );

    $blogs[ $special_blog_id ] = (object) array(
    'userblog_id' => $special_blog_id,
    'blogname'    => $new_blog->blogname,
    'domain'      => $new_blog->domain,
    'path'        => $new_blog->path,
    'site_id'     => $new_blog->site_id,
    'siteurl'     => $new_blog->siteurl,
    'archived'    => 0,
    'spam'        => 0,
    'deleted'     => 0
    );

    return $blogs;
}