Call to undefined function get_user_meta() – trying to access data in MySQL from custom fields

For one thing, you don’t have to include wp-config or wp-db when you use wp-load. And as t31os says, single-quotes around variables will keep them from being interpreted by PHP.

<?php
include_once('wp-load.php');
?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<title>Untitled Document</title>
<style type="text/css">
/* styles go here */
</style>
</head>

<body>
<div id="wrapper">

<div id="top_badge">

<?php 

if ( is_user_logged_in() ) {

global $user_ID;

// Get the 'paypal_user' info for the user. Just fetch a single
// value, not an array.
$paypal_user = get_user_meta($user_ID, 'paypal_user', true);
$exp_date = $paypal_user->expire_date;
?>

<div id="member_no_1"><?php echo $user_ID ?></div>
<div id="member_lp_1">PR34 2PL</div>
<div id="member_exp_1"><?php echo $exp_date ?></div>

</div>
<?php } ?>
</div>

</body>
</html>

Give that a shot. Works for me, assuming that you put this file in the same directory that wp-load.php lives in.