WordPress redirect to splash page once a day for the first week

This can be achieved with Two Cookie.

1. Saves the unix time of user visits first time. Expiration time set to a higher value, A cookie that never expires.
2. Set another cookie that is valid for 24 hours.

Code

function wpse144762_check_accessed(){

    if( !isset($_COOKIE['first_time']) ) {
        // user visiting for the first time
        setcookie('first_time', 'yes', time() + 60 * 60 * 24 * 365 * 10); // expires in 10 years 
        define("ACCESSED", false);
    }elseif ( 
                !isset($_COOKIE['accessed']) 
                && intval($_COOKIE['first_time']) > time() - 60 * 60 * 24 * 7 
    ) { 
        setcookie('accessed', 'yes', time() + 3600*24); 
        define("ACCESSED", false);
    }else{
        define("ACCESSED", true);
    }
}
add_action("template_redirect", "wpse144762_check_accessed");

Code is not tested. But this should give you the idea