Admin first hook that outputs HTML?

You can try this one:

/**
 * Fires as an admin screen or script is being initialized.
 *
 * Note, this does not just run on user-facing admin screens.
 * It runs on admin-ajax.php and admin-post.php as well.
 *  
 * This is roughly analgous to the more general 'init' hook, which fires earlier.
 *
 * @since 2.5.0
 */
do_action( 'admin_init' );

if you need a hook activated on every admin page.

For example:

add_action( 'admin_init', 'wpse_admin_init' );

function wpse_admin_init( $buffer )
{
    if( ! defined( 'DOING_AJAX') || ! DOING_AJAX )
        ob_start( 'wpse_buffering' );
}

function wpse_buffering( $buffer )
{       
    return $buffer;
}

ps: If you need to hook into the part after </html>, you can use the PHP function register_shutdown_function() to run your own callback after the PHP script has finished executing. But notice that then the output buffers have already be sent to the client, according to this comment on the PHP docs.