mysql_connect("nope.com","nope","nope") or die ("Verbindung zur Datenbank konnte nicht hergestellt werden");
mysql_select_db("nope") or die ("Datenbank konnte nicht verbunden werden");
$nnamenow = $_SESSION['username'];
echo(mysql_fetch_object(mysql_query("SELECT Name FROM USER WHERE Username LIKE '$nnamenow'")));

$result = mysql_query("SELECT Name FROM USER WHERE Username LIKE '$nnamenow'");

while($row = mysql_fetch_assoc($result)) {
    echo $row['Name'];
}

$obj = mysql_fetch_object(...);
echo $obj->Name;

$result = mysql_fetch_object(mysql_query("SELECT Name FROM USER WHERE Username LIKE '$nnamenow'"));
var_dump($result); // echo $result->Name;